{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r03-0001","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-descent"],"title_zh":"梯度下降","stem_zh":"standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-quadratic-gradient；明示資料={\"diagonal\":[1,2],\"linear\":[-1,-2],\"point\":[-2,-3]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([1, 2], [-1, -2], [-2, -3], [-3, -8], 19, fixed_standard_Euclidean_coordinate_order)","explanation_zh":"Wave6第十三層核對=([1, 2], [-1, -2], [-2, -3], [-3, -8], 19, fixed_standard_Euclidean_coordinate_order)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["梯度下降"]},"variant":{"family_id":"qf-mod50-gd-quadratic-gradient","seed":202611193001,"parameters":{"diagonal":[1,2],"linear":[-1,-2],"point":[-2,-3],"domain":"Exact_diagonal_quadratic_gradient_descent","operation":"gd-quadratic-gradient","semantic_template":"gd-quadratic-gradient|{\"diagonal\":[[1,1],[2,1]],\"linear\":[[-1,1],[-2,1]],\"point\":[[-2,1],[-3,1]]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r03-0002","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-descent"],"title_zh":"梯度下降","stem_zh":"standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-quadratic-gradient；明示資料={\"diagonal\":[2,3],\"linear\":[0,-1],\"point\":[-1,-2]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([2, 3], [0, -1], [-1, -2], [-2, -7], 9, fixed_standard_Euclidean_coordinate_order)","explanation_zh":"Wave6第十三層核對=([2, 3], [0, -1], [-1, -2], [-2, -7], 9, fixed_standard_Euclidean_coordinate_order)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["梯度下降"]},"variant":{"family_id":"qf-mod50-gd-quadratic-gradient","seed":202611193002,"parameters":{"diagonal":[2,3],"linear":[0,-1],"point":[-1,-2],"domain":"Exact_diagonal_quadratic_gradient_descent","operation":"gd-quadratic-gradient","semantic_template":"gd-quadratic-gradient|{\"diagonal\":[[2,1],[3,1]],\"linear\":[[0,1],[-1,1]],\"point\":[[-1,1],[-2,1]]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r03-0003","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-descent"],"title_zh":"梯度下降","stem_zh":"standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-quadratic-gradient；明示資料={\"diagonal\":[3,4],\"linear\":[1,0],\"point\":[0,-1]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([3, 4], [1, 0], [0, -1], [1, -4], 2, fixed_standard_Euclidean_coordinate_order)","explanation_zh":"Wave6第十三層核對=([3, 4], [1, 0], [0, -1], [1, -4], 2, fixed_standard_Euclidean_coordinate_order)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["梯度下降"]},"variant":{"family_id":"qf-mod50-gd-quadratic-gradient","seed":202611193003,"parameters":{"diagonal":[3,4],"linear":[1,0],"point":[0,-1],"domain":"Exact_diagonal_quadratic_gradient_descent","operation":"gd-quadratic-gradient","semantic_template":"gd-quadratic-gradient|{\"diagonal\":[[3,1],[4,1]],\"linear\":[[1,1],[0,1]],\"point\":[[0,1],[-1,1]]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r03-0004","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-descent"],"title_zh":"梯度下降","stem_zh":"standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-quadratic-gradient；明示資料={\"diagonal\":[1,5],\"linear\":[2,1],\"point\":[1,0]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([1, 5], [2, 1], [1, 0], [3, 1], 5/2, fixed_standard_Euclidean_coordinate_order)","explanation_zh":"Wave6第十三層核對=([1, 5], [2, 1], [1, 0], [3, 1], 5/2, fixed_standard_Euclidean_coordinate_order)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["梯度下降"]},"variant":{"family_id":"qf-mod50-gd-quadratic-gradient","seed":202611193004,"parameters":{"diagonal":[1,5],"linear":[2,1],"point":[1,0],"domain":"Exact_diagonal_quadratic_gradient_descent","operation":"gd-quadratic-gradient","semantic_template":"gd-quadratic-gradient|{\"diagonal\":[[1,1],[5,1]],\"linear\":[[2,1],[1,1]],\"point\":[[1,1],[0,1]]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r03-0005","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-descent"],"title_zh":"梯度下降","stem_zh":"standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-quadratic-gradient；明示資料={\"diagonal\":[2,2],\"linear\":[-2,2],\"point\":[2,1]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([2, 2], [-2, 2], [2, 1], [2, 4], 3, fixed_standard_Euclidean_coordinate_order)","explanation_zh":"Wave6第十三層核對=([2, 2], [-2, 2], [2, 1], [2, 4], 3, fixed_standard_Euclidean_coordinate_order)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["梯度下降"]},"variant":{"family_id":"qf-mod50-gd-quadratic-gradient","seed":202611193005,"parameters":{"diagonal":[2,2],"linear":[-2,2],"point":[2,1],"domain":"Exact_diagonal_quadratic_gradient_descent","operation":"gd-quadratic-gradient","semantic_template":"gd-quadratic-gradient|{\"diagonal\":[[2,1],[2,1]],\"linear\":[[-2,1],[2,1]],\"point\":[[2,1],[1,1]]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r03-0006","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-descent"],"title_zh":"梯度下降","stem_zh":"standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-quadratic-gradient；明示資料={\"diagonal\":[3,3],\"linear\":[-1,3],\"point\":[-2,2]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([3, 3], [-1, 3], [-2, 2], [-7, 9], 20, fixed_standard_Euclidean_coordinate_order)","explanation_zh":"Wave6第十三層核對=([3, 3], [-1, 3], [-2, 2], [-7, 9], 20, fixed_standard_Euclidean_coordinate_order)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["梯度下降"]},"variant":{"family_id":"qf-mod50-gd-quadratic-gradient","seed":202611193006,"parameters":{"diagonal":[3,3],"linear":[-1,3],"point":[-2,2],"domain":"Exact_diagonal_quadratic_gradient_descent","operation":"gd-quadratic-gradient","semantic_template":"gd-quadratic-gradient|{\"diagonal\":[[3,1],[3,1]],\"linear\":[[-1,1],[3,1]],\"point\":[[-2,1],[2,1]]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r03-0007","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-descent"],"title_zh":"梯度下降","stem_zh":"standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-quadratic-gradient；明示資料={\"diagonal\":[1,4],\"linear\":[0,-3],\"point\":[-1,3]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([1, 4], [0, -3], [-1, 3], [-1, 9], 19/2, fixed_standard_Euclidean_coordinate_order)","explanation_zh":"Wave6第十三層核對=([1, 4], [0, -3], [-1, 3], [-1, 9], 19/2, fixed_standard_Euclidean_coordinate_order)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["梯度下降"]},"variant":{"family_id":"qf-mod50-gd-quadratic-gradient","seed":202611193007,"parameters":{"diagonal":[1,4],"linear":[0,-3],"point":[-1,3],"domain":"Exact_diagonal_quadratic_gradient_descent","operation":"gd-quadratic-gradient","semantic_template":"gd-quadratic-gradient|{\"diagonal\":[[1,1],[4,1]],\"linear\":[[0,1],[-3,1]],\"point\":[[-1,1],[3,1]]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r03-0008","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-descent"],"title_zh":"梯度下降","stem_zh":"standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-quadratic-gradient；明示資料={\"diagonal\":[2,5],\"linear\":[1,-2],\"point\":[0,-3]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([2, 5], [1, -2], [0, -3], [1, -17], 57/2, fixed_standard_Euclidean_coordinate_order)","explanation_zh":"Wave6第十三層核對=([2, 5], [1, -2], [0, -3], [1, -17], 57/2, fixed_standard_Euclidean_coordinate_order)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["梯度下降"]},"variant":{"family_id":"qf-mod50-gd-quadratic-gradient","seed":202611193008,"parameters":{"diagonal":[2,5],"linear":[1,-2],"point":[0,-3],"domain":"Exact_diagonal_quadratic_gradient_descent","operation":"gd-quadratic-gradient","semantic_template":"gd-quadratic-gradient|{\"diagonal\":[[2,1],[5,1]],\"linear\":[[1,1],[-2,1]],\"point\":[[0,1],[-3,1]]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r03-0009","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-descent"],"title_zh":"梯度下降","stem_zh":"standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-quadratic-gradient；明示資料={\"diagonal\":[3,2],\"linear\":[2,-1],\"point\":[1,-2]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([3, 2], [2, -1], [1, -2], [5, -5], 19/2, fixed_standard_Euclidean_coordinate_order)","explanation_zh":"Wave6第十三層核對=([3, 2], [2, -1], [1, -2], [5, -5], 19/2, fixed_standard_Euclidean_coordinate_order)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度下降"]},"variant":{"family_id":"qf-mod50-gd-quadratic-gradient","seed":202611193009,"parameters":{"diagonal":[3,2],"linear":[2,-1],"point":[1,-2],"domain":"Exact_diagonal_quadratic_gradient_descent","operation":"gd-quadratic-gradient","semantic_template":"gd-quadratic-gradient|{\"diagonal\":[[3,1],[2,1]],\"linear\":[[2,1],[-1,1]],\"point\":[[1,1],[-2,1]]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r03-0010","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-descent"],"title_zh":"梯度下降","stem_zh":"standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-quadratic-gradient；明示資料={\"diagonal\":[1,3],\"linear\":[-2,0],\"point\":[2,-1]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([1, 3], [-2, 0], [2, -1], [0, -3], -1/2, fixed_standard_Euclidean_coordinate_order)","explanation_zh":"Wave6第十三層核對=([1, 3], [-2, 0], [2, -1], [0, -3], -1/2, fixed_standard_Euclidean_coordinate_order)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度下降"]},"variant":{"family_id":"qf-mod50-gd-quadratic-gradient","seed":202611193010,"parameters":{"diagonal":[1,3],"linear":[-2,0],"point":[2,-1],"domain":"Exact_diagonal_quadratic_gradient_descent","operation":"gd-quadratic-gradient","semantic_template":"gd-quadratic-gradient|{\"diagonal\":[[1,1],[3,1]],\"linear\":[[-2,1],[0,1]],\"point\":[[2,1],[-1,1]]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r03-0011","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-descent"],"title_zh":"梯度下降","stem_zh":"standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-quadratic-gradient；明示資料={\"diagonal\":[2,4],\"linear\":[-1,1],\"point\":[-2,0]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([2, 4], [-1, 1], [-2, 0], [-5, 1], 6, fixed_standard_Euclidean_coordinate_order)","explanation_zh":"Wave6第十三層核對=([2, 4], [-1, 1], [-2, 0], [-5, 1], 6, fixed_standard_Euclidean_coordinate_order)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度下降"]},"variant":{"family_id":"qf-mod50-gd-quadratic-gradient","seed":202611193011,"parameters":{"diagonal":[2,4],"linear":[-1,1],"point":[-2,0],"domain":"Exact_diagonal_quadratic_gradient_descent","operation":"gd-quadratic-gradient","semantic_template":"gd-quadratic-gradient|{\"diagonal\":[[2,1],[4,1]],\"linear\":[[-1,1],[1,1]],\"point\":[[-2,1],[0,1]]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r03-0012","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-descent"],"title_zh":"梯度下降","stem_zh":"standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-quadratic-gradient；明示資料={\"diagonal\":[3,5],\"linear\":[0,2],\"point\":[-1,1]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([3, 5], [0, 2], [-1, 1], [-3, 7], 6, fixed_standard_Euclidean_coordinate_order)","explanation_zh":"Wave6第十三層核對=([3, 5], [0, 2], [-1, 1], [-3, 7], 6, fixed_standard_Euclidean_coordinate_order)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度下降"]},"variant":{"family_id":"qf-mod50-gd-quadratic-gradient","seed":202611193012,"parameters":{"diagonal":[3,5],"linear":[0,2],"point":[-1,1],"domain":"Exact_diagonal_quadratic_gradient_descent","operation":"gd-quadratic-gradient","semantic_template":"gd-quadratic-gradient|{\"diagonal\":[[3,1],[5,1]],\"linear\":[[0,1],[2,1]],\"point\":[[-1,1],[1,1]]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r03-0013","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-descent"],"title_zh":"梯度下降","stem_zh":"standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-quadratic-gradient；明示資料={\"diagonal\":[1,2],\"linear\":[1,3],\"point\":[0,2]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([1, 2], [1, 3], [0, 2], [1, 7], 10, fixed_standard_Euclidean_coordinate_order)","explanation_zh":"Wave6第十三層核對=([1, 2], [1, 3], [0, 2], [1, 7], 10, fixed_standard_Euclidean_coordinate_order)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度下降"]},"variant":{"family_id":"qf-mod50-gd-quadratic-gradient","seed":202611193013,"parameters":{"diagonal":[1,2],"linear":[1,3],"point":[0,2],"domain":"Exact_diagonal_quadratic_gradient_descent","operation":"gd-quadratic-gradient","semantic_template":"gd-quadratic-gradient|{\"diagonal\":[[1,1],[2,1]],\"linear\":[[1,1],[3,1]],\"point\":[[0,1],[2,1]]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r03-0014","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-descent"],"title_zh":"梯度下降","stem_zh":"standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-quadratic-gradient；明示資料={\"diagonal\":[2,3],\"linear\":[2,-3],\"point\":[1,3]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([2, 3], [2, -3], [1, 3], [4, 6], 15/2, fixed_standard_Euclidean_coordinate_order)","explanation_zh":"Wave6第十三層核對=([2, 3], [2, -3], [1, 3], [4, 6], 15/2, fixed_standard_Euclidean_coordinate_order)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度下降"]},"variant":{"family_id":"qf-mod50-gd-quadratic-gradient","seed":202611193014,"parameters":{"diagonal":[2,3],"linear":[2,-3],"point":[1,3],"domain":"Exact_diagonal_quadratic_gradient_descent","operation":"gd-quadratic-gradient","semantic_template":"gd-quadratic-gradient|{\"diagonal\":[[2,1],[3,1]],\"linear\":[[2,1],[-3,1]],\"point\":[[1,1],[3,1]]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r03-0015","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-descent"],"title_zh":"梯度下降","stem_zh":"standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-quadratic-gradient；明示資料={\"diagonal\":[3,4],\"linear\":[-2,-2],\"point\":[2,-3]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([3, 4], [-2, -2], [2, -3], [4, -14], 26, fixed_standard_Euclidean_coordinate_order)","explanation_zh":"Wave6第十三層核對=([3, 4], [-2, -2], [2, -3], [4, -14], 26, fixed_standard_Euclidean_coordinate_order)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度下降"]},"variant":{"family_id":"qf-mod50-gd-quadratic-gradient","seed":202611193015,"parameters":{"diagonal":[3,4],"linear":[-2,-2],"point":[2,-3],"domain":"Exact_diagonal_quadratic_gradient_descent","operation":"gd-quadratic-gradient","semantic_template":"gd-quadratic-gradient|{\"diagonal\":[[3,1],[4,1]],\"linear\":[[-2,1],[-2,1]],\"point\":[[2,1],[-3,1]]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r03-0016","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-descent"],"title_zh":"梯度下降","stem_zh":"standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-quadratic-gradient；明示資料={\"diagonal\":[1,5],\"linear\":[-1,-1],\"point\":[-2,-2]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([1, 5], [-1, -1], [-2, -2], [-3, -11], 16, fixed_standard_Euclidean_coordinate_order)","explanation_zh":"Wave6第十三層核對=([1, 5], [-1, -1], [-2, -2], [-3, -11], 16, fixed_standard_Euclidean_coordinate_order)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度下降"]},"variant":{"family_id":"qf-mod50-gd-quadratic-gradient","seed":202611193016,"parameters":{"diagonal":[1,5],"linear":[-1,-1],"point":[-2,-2],"domain":"Exact_diagonal_quadratic_gradient_descent","operation":"gd-quadratic-gradient","semantic_template":"gd-quadratic-gradient|{\"diagonal\":[[1,1],[5,1]],\"linear\":[[-1,1],[-1,1]],\"point\":[[-2,1],[-2,1]]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r03-0017","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-descent"],"title_zh":"梯度下降","stem_zh":"standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-quadratic-gradient；明示資料={\"diagonal\":[2,2],\"linear\":[0,0],\"point\":[-1,-1]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([2, 2], [0, 0], [-1, -1], [-2, -2], 2, fixed_standard_Euclidean_coordinate_order)","explanation_zh":"Wave6第十三層核對=([2, 2], [0, 0], [-1, -1], [-2, -2], 2, fixed_standard_Euclidean_coordinate_order)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度下降"]},"variant":{"family_id":"qf-mod50-gd-quadratic-gradient","seed":202611193017,"parameters":{"diagonal":[2,2],"linear":[0,0],"point":[-1,-1],"domain":"Exact_diagonal_quadratic_gradient_descent","operation":"gd-quadratic-gradient","semantic_template":"gd-quadratic-gradient|{\"diagonal\":[[2,1],[2,1]],\"linear\":[[0,1],[0,1]],\"point\":[[-1,1],[-1,1]]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r03-0018","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-descent"],"title_zh":"梯度下降","stem_zh":"standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-quadratic-gradient；明示資料={\"diagonal\":[3,3],\"linear\":[1,1],\"point\":[0,0]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([3, 3], [1, 1], [0, 0], [1, 1], 0, fixed_standard_Euclidean_coordinate_order)","explanation_zh":"Wave6第十三層核對=([3, 3], [1, 1], [0, 0], [1, 1], 0, fixed_standard_Euclidean_coordinate_order)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["梯度下降"]},"variant":{"family_id":"qf-mod50-gd-quadratic-gradient","seed":202611193018,"parameters":{"diagonal":[3,3],"linear":[1,1],"point":[0,0],"domain":"Exact_diagonal_quadratic_gradient_descent","operation":"gd-quadratic-gradient","semantic_template":"gd-quadratic-gradient|{\"diagonal\":[[3,1],[3,1]],\"linear\":[[1,1],[1,1]],\"point\":[[0,1],[0,1]]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r03-0019","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-descent"],"title_zh":"梯度下降","stem_zh":"standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-quadratic-gradient；明示資料={\"diagonal\":[1,4],\"linear\":[2,2],\"point\":[1,1]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([1, 4], [2, 2], [1, 1], [3, 6], 13/2, fixed_standard_Euclidean_coordinate_order)","explanation_zh":"Wave6第十三層核對=([1, 4], [2, 2], [1, 1], [3, 6], 13/2, fixed_standard_Euclidean_coordinate_order)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["梯度下降"]},"variant":{"family_id":"qf-mod50-gd-quadratic-gradient","seed":202611193019,"parameters":{"diagonal":[1,4],"linear":[2,2],"point":[1,1],"domain":"Exact_diagonal_quadratic_gradient_descent","operation":"gd-quadratic-gradient","semantic_template":"gd-quadratic-gradient|{\"diagonal\":[[1,1],[4,1]],\"linear\":[[2,1],[2,1]],\"point\":[[1,1],[1,1]]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r03-0020","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-descent"],"title_zh":"梯度下降","stem_zh":"standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-quadratic-gradient；明示資料={\"diagonal\":[2,5],\"linear\":[-2,3],\"point\":[2,2]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([2, 5], [-2, 3], [2, 2], [2, 13], 16, fixed_standard_Euclidean_coordinate_order)","explanation_zh":"Wave6第十三層核對=([2, 5], [-2, 3], [2, 2], [2, 13], 16, fixed_standard_Euclidean_coordinate_order)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["梯度下降"]},"variant":{"family_id":"qf-mod50-gd-quadratic-gradient","seed":202611193020,"parameters":{"diagonal":[2,5],"linear":[-2,3],"point":[2,2],"domain":"Exact_diagonal_quadratic_gradient_descent","operation":"gd-quadratic-gradient","semantic_template":"gd-quadratic-gradient|{\"diagonal\":[[2,1],[5,1]],\"linear\":[[-2,1],[3,1]],\"point\":[[2,1],[2,1]]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r03-0021","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-descent"],"title_zh":"梯度下降","stem_zh":"standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-quadratic-gradient；明示資料={\"diagonal\":[3,2],\"linear\":[-1,-3],\"point\":[-2,3]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([3, 2], [-1, -3], [-2, 3], [-7, 3], 8, fixed_standard_Euclidean_coordinate_order)","explanation_zh":"Wave6第十三層核對=([3, 2], [-1, -3], [-2, 3], [-7, 3], 8, fixed_standard_Euclidean_coordinate_order)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["梯度下降"]},"variant":{"family_id":"qf-mod50-gd-quadratic-gradient","seed":202611193021,"parameters":{"diagonal":[3,2],"linear":[-1,-3],"point":[-2,3],"domain":"Exact_diagonal_quadratic_gradient_descent","operation":"gd-quadratic-gradient","semantic_template":"gd-quadratic-gradient|{\"diagonal\":[[3,1],[2,1]],\"linear\":[[-1,1],[-3,1]],\"point\":[[-2,1],[3,1]]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r03-0022","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-descent"],"title_zh":"梯度下降","stem_zh":"standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-quadratic-gradient；明示資料={\"diagonal\":[1,3],\"linear\":[0,-2],\"point\":[-1,-3]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([1, 3], [0, -2], [-1, -3], [-1, -11], 20, fixed_standard_Euclidean_coordinate_order)","explanation_zh":"Wave6第十三層核對=([1, 3], [0, -2], [-1, -3], [-1, -11], 20, fixed_standard_Euclidean_coordinate_order)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["梯度下降"]},"variant":{"family_id":"qf-mod50-gd-quadratic-gradient","seed":202611193022,"parameters":{"diagonal":[1,3],"linear":[0,-2],"point":[-1,-3],"domain":"Exact_diagonal_quadratic_gradient_descent","operation":"gd-quadratic-gradient","semantic_template":"gd-quadratic-gradient|{\"diagonal\":[[1,1],[3,1]],\"linear\":[[0,1],[-2,1]],\"point\":[[-1,1],[-3,1]]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r03-0023","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-descent"],"title_zh":"梯度下降","stem_zh":"standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-quadratic-gradient；明示資料={\"diagonal\":[2,4],\"linear\":[1,-1],\"point\":[0,-2]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([2, 4], [1, -1], [0, -2], [1, -9], 10, fixed_standard_Euclidean_coordinate_order)","explanation_zh":"Wave6第十三層核對=([2, 4], [1, -1], [0, -2], [1, -9], 10, fixed_standard_Euclidean_coordinate_order)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["梯度下降"]},"variant":{"family_id":"qf-mod50-gd-quadratic-gradient","seed":202611193023,"parameters":{"diagonal":[2,4],"linear":[1,-1],"point":[0,-2],"domain":"Exact_diagonal_quadratic_gradient_descent","operation":"gd-quadratic-gradient","semantic_template":"gd-quadratic-gradient|{\"diagonal\":[[2,1],[4,1]],\"linear\":[[1,1],[-1,1]],\"point\":[[0,1],[-2,1]]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r03-0024","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-descent"],"title_zh":"梯度下降","stem_zh":"standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-quadratic-gradient；明示資料={\"diagonal\":[3,5],\"linear\":[2,0],\"point\":[1,-1]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([3, 5], [2, 0], [1, -1], [5, -5], 6, fixed_standard_Euclidean_coordinate_order)","explanation_zh":"Wave6第十三層核對=([3, 5], [2, 0], [1, -1], [5, -5], 6, fixed_standard_Euclidean_coordinate_order)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["梯度下降"]},"variant":{"family_id":"qf-mod50-gd-quadratic-gradient","seed":202611193024,"parameters":{"diagonal":[3,5],"linear":[2,0],"point":[1,-1],"domain":"Exact_diagonal_quadratic_gradient_descent","operation":"gd-quadratic-gradient","semantic_template":"gd-quadratic-gradient|{\"diagonal\":[[3,1],[5,1]],\"linear\":[[2,1],[0,1]],\"point\":[[1,1],[-1,1]]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r03-0025","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-descent"],"title_zh":"梯度下降","stem_zh":"standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-quadratic-gradient；明示資料={\"diagonal\":[1,2],\"linear\":[-2,1],\"point\":[2,0]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([1, 2], [-2, 1], [2, 0], [0, 1], -2, fixed_standard_Euclidean_coordinate_order)","explanation_zh":"Wave6第十三層核對=([1, 2], [-2, 1], [2, 0], [0, 1], -2, fixed_standard_Euclidean_coordinate_order)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["梯度下降"]},"variant":{"family_id":"qf-mod50-gd-quadratic-gradient","seed":202611193025,"parameters":{"diagonal":[1,2],"linear":[-2,1],"point":[2,0],"domain":"Exact_diagonal_quadratic_gradient_descent","operation":"gd-quadratic-gradient","semantic_template":"gd-quadratic-gradient|{\"diagonal\":[[1,1],[2,1]],\"linear\":[[-2,1],[1,1]],\"point\":[[2,1],[0,1]]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r03-0026","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-descent"],"title_zh":"梯度下降","stem_zh":"standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-one-step；明示資料={\"diagonal\":[1,2],\"linear\":[-1,-2],\"point\":[-2,-3],\"step_size\":[1,4]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([-2, -3], [-3, -8], 1/4, [-5/4, -1], 19, 161/32, 是, positive_step_alone_does_not_guarantee_descent)","explanation_zh":"Wave6第十三層核對=([-2, -3], [-3, -8], 1/4, [-5/4, -1], 19, 161/32, 是, positive_step_alone_does_not_guarantee_descent)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["梯度下降"]},"variant":{"family_id":"qf-mod50-gd-one-step","seed":202611193026,"parameters":{"diagonal":[1,2],"linear":[-1,-2],"point":[-2,-3],"step_size":[1,4],"domain":"Exact_diagonal_quadratic_gradient_descent","operation":"gd-one-step","semantic_template":"gd-one-step|{\"diagonal\":[[1,1],[2,1]],\"linear\":[[-1,1],[-2,1]],\"point\":[[-2,1],[-3,1]],\"step_size\":[1,4]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r03-0027","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-descent"],"title_zh":"梯度下降","stem_zh":"standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-one-step；明示資料={\"diagonal\":[2,3],\"linear\":[0,-1],\"point\":[-1,-2],\"step_size\":[1,4]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([-1, -2], [-2, -7], 1/4, [-1/2, -1/4], 9, 19/32, 是, positive_step_alone_does_not_guarantee_descent)","explanation_zh":"Wave6第十三層核對=([-1, -2], [-2, -7], 1/4, [-1/2, -1/4], 9, 19/32, 是, positive_step_alone_does_not_guarantee_descent)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["梯度下降"]},"variant":{"family_id":"qf-mod50-gd-one-step","seed":202611193027,"parameters":{"diagonal":[2,3],"linear":[0,-1],"point":[-1,-2],"step_size":[1,4],"domain":"Exact_diagonal_quadratic_gradient_descent","operation":"gd-one-step","semantic_template":"gd-one-step|{\"diagonal\":[[2,1],[3,1]],\"linear\":[[0,1],[-1,1]],\"point\":[[-1,1],[-2,1]],\"step_size\":[1,4]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r03-0028","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-descent"],"title_zh":"梯度下降","stem_zh":"standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-one-step；明示資料={\"diagonal\":[3,4],\"linear\":[1,0],\"point\":[0,-1],\"step_size\":[1,4]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([0, -1], [1, -4], 1/4, [-1/4, 0], 2, -5/32, 是, positive_step_alone_does_not_guarantee_descent)","explanation_zh":"Wave6第十三層核對=([0, -1], [1, -4], 1/4, [-1/4, 0], 2, -5/32, 是, positive_step_alone_does_not_guarantee_descent)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["梯度下降"]},"variant":{"family_id":"qf-mod50-gd-one-step","seed":202611193028,"parameters":{"diagonal":[3,4],"linear":[1,0],"point":[0,-1],"step_size":[1,4],"domain":"Exact_diagonal_quadratic_gradient_descent","operation":"gd-one-step","semantic_template":"gd-one-step|{\"diagonal\":[[3,1],[4,1]],\"linear\":[[1,1],[0,1]],\"point\":[[0,1],[-1,1]],\"step_size\":[1,4]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r03-0029","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-descent"],"title_zh":"梯度下降","stem_zh":"standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-one-step；明示資料={\"diagonal\":[1,5],\"linear\":[2,1],\"point\":[1,0],\"step_size\":[1,4]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([1, 0], [3, 1], 1/4, [1/4, -1/4], 5/2, 7/16, 是, positive_step_alone_does_not_guarantee_descent)","explanation_zh":"Wave6第十三層核對=([1, 0], [3, 1], 1/4, [1/4, -1/4], 5/2, 7/16, 是, positive_step_alone_does_not_guarantee_descent)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["梯度下降"]},"variant":{"family_id":"qf-mod50-gd-one-step","seed":202611193029,"parameters":{"diagonal":[1,5],"linear":[2,1],"point":[1,0],"step_size":[1,4],"domain":"Exact_diagonal_quadratic_gradient_descent","operation":"gd-one-step","semantic_template":"gd-one-step|{\"diagonal\":[[1,1],[5,1]],\"linear\":[[2,1],[1,1]],\"point\":[[1,1],[0,1]],\"step_size\":[1,4]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r03-0030","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-descent"],"title_zh":"梯度下降","stem_zh":"standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-one-step；明示資料={\"diagonal\":[2,2],\"linear\":[-2,2],\"point\":[2,1],\"step_size\":[1,4]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([2, 1], [2, 4], 1/4, [3/2, 0], 3, -3/4, 是, positive_step_alone_does_not_guarantee_descent)","explanation_zh":"Wave6第十三層核對=([2, 1], [2, 4], 1/4, [3/2, 0], 3, -3/4, 是, positive_step_alone_does_not_guarantee_descent)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["梯度下降"]},"variant":{"family_id":"qf-mod50-gd-one-step","seed":202611193030,"parameters":{"diagonal":[2,2],"linear":[-2,2],"point":[2,1],"step_size":[1,4],"domain":"Exact_diagonal_quadratic_gradient_descent","operation":"gd-one-step","semantic_template":"gd-one-step|{\"diagonal\":[[2,1],[2,1]],\"linear\":[[-2,1],[2,1]],\"point\":[[2,1],[1,1]],\"step_size\":[1,4]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r03-0031","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-descent"],"title_zh":"梯度下降","stem_zh":"standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-one-step；明示資料={\"diagonal\":[3,3],\"linear\":[-1,3],\"point\":[-2,2],\"step_size\":[1,4]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([-2, 2], [-7, 9], 1/4, [-1/4, -1/4], 20, -5/16, 是, positive_step_alone_does_not_guarantee_descent)","explanation_zh":"Wave6第十三層核對=([-2, 2], [-7, 9], 1/4, [-1/4, -1/4], 20, -5/16, 是, positive_step_alone_does_not_guarantee_descent)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["梯度下降"]},"variant":{"family_id":"qf-mod50-gd-one-step","seed":202611193031,"parameters":{"diagonal":[3,3],"linear":[-1,3],"point":[-2,2],"step_size":[1,4],"domain":"Exact_diagonal_quadratic_gradient_descent","operation":"gd-one-step","semantic_template":"gd-one-step|{\"diagonal\":[[3,1],[3,1]],\"linear\":[[-1,1],[3,1]],\"point\":[[-2,1],[2,1]],\"step_size\":[1,4]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r03-0032","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-descent"],"title_zh":"梯度下降","stem_zh":"standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-one-step；明示資料={\"diagonal\":[1,4],\"linear\":[0,-3],\"point\":[-1,3],\"step_size\":[1,4]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([-1, 3], [-1, 9], 1/4, [-3/4, 3/4], 19/2, -27/32, 是, positive_step_alone_does_not_guarantee_descent)","explanation_zh":"Wave6第十三層核對=([-1, 3], [-1, 9], 1/4, [-3/4, 3/4], 19/2, -27/32, 是, positive_step_alone_does_not_guarantee_descent)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["梯度下降"]},"variant":{"family_id":"qf-mod50-gd-one-step","seed":202611193032,"parameters":{"diagonal":[1,4],"linear":[0,-3],"point":[-1,3],"step_size":[1,4],"domain":"Exact_diagonal_quadratic_gradient_descent","operation":"gd-one-step","semantic_template":"gd-one-step|{\"diagonal\":[[1,1],[4,1]],\"linear\":[[0,1],[-3,1]],\"point\":[[-1,1],[3,1]],\"step_size\":[1,4]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r03-0033","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-descent"],"title_zh":"梯度下降","stem_zh":"standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-one-step；明示資料={\"diagonal\":[2,5],\"linear\":[1,-2],\"point\":[0,-3],\"step_size\":[1,4]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([0, -3], [1, -17], 1/4, [-1/4, 5/4], 57/2, 39/32, 是, positive_step_alone_does_not_guarantee_descent)","explanation_zh":"Wave6第十三層核對=([0, -3], [1, -17], 1/4, [-1/4, 5/4], 57/2, 39/32, 是, positive_step_alone_does_not_guarantee_descent)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["梯度下降"]},"variant":{"family_id":"qf-mod50-gd-one-step","seed":202611193033,"parameters":{"diagonal":[2,5],"linear":[1,-2],"point":[0,-3],"step_size":[1,4],"domain":"Exact_diagonal_quadratic_gradient_descent","operation":"gd-one-step","semantic_template":"gd-one-step|{\"diagonal\":[[2,1],[5,1]],\"linear\":[[1,1],[-2,1]],\"point\":[[0,1],[-3,1]],\"step_size\":[1,4]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r03-0034","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-descent"],"title_zh":"梯度下降","stem_zh":"standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-one-step；明示資料={\"diagonal\":[3,2],\"linear\":[2,-1],\"point\":[1,-2],\"step_size\":[1,4]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([1, -2], [5, -5], 1/4, [-1/4, -3/4], 19/2, 29/32, 是, positive_step_alone_does_not_guarantee_descent)","explanation_zh":"Wave6第十三層核對=([1, -2], [5, -5], 1/4, [-1/4, -3/4], 19/2, 29/32, 是, positive_step_alone_does_not_guarantee_descent)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度下降"]},"variant":{"family_id":"qf-mod50-gd-one-step","seed":202611193034,"parameters":{"diagonal":[3,2],"linear":[2,-1],"point":[1,-2],"step_size":[1,4],"domain":"Exact_diagonal_quadratic_gradient_descent","operation":"gd-one-step","semantic_template":"gd-one-step|{\"diagonal\":[[3,1],[2,1]],\"linear\":[[2,1],[-1,1]],\"point\":[[1,1],[-2,1]],\"step_size\":[1,4]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r03-0035","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-descent"],"title_zh":"梯度下降","stem_zh":"standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-one-step；明示資料={\"diagonal\":[1,3],\"linear\":[-2,0],\"point\":[2,-1],\"step_size\":[1,4]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([2, -1], [0, -3], 1/4, [2, -1/4], -1/2, -61/32, 是, positive_step_alone_does_not_guarantee_descent)","explanation_zh":"Wave6第十三層核對=([2, -1], [0, -3], 1/4, [2, -1/4], -1/2, -61/32, 是, positive_step_alone_does_not_guarantee_descent)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度下降"]},"variant":{"family_id":"qf-mod50-gd-one-step","seed":202611193035,"parameters":{"diagonal":[1,3],"linear":[-2,0],"point":[2,-1],"step_size":[1,4],"domain":"Exact_diagonal_quadratic_gradient_descent","operation":"gd-one-step","semantic_template":"gd-one-step|{\"diagonal\":[[1,1],[3,1]],\"linear\":[[-2,1],[0,1]],\"point\":[[2,1],[-1,1]],\"step_size\":[1,4]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r03-0036","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-descent"],"title_zh":"梯度下降","stem_zh":"standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-one-step；明示資料={\"diagonal\":[2,4],\"linear\":[-1,1],\"point\":[-2,0],\"step_size\":[1,4]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([-2, 0], [-5, 1], 1/4, [-3/4, -1/4], 6, 19/16, 是, positive_step_alone_does_not_guarantee_descent)","explanation_zh":"Wave6第十三層核對=([-2, 0], [-5, 1], 1/4, [-3/4, -1/4], 6, 19/16, 是, positive_step_alone_does_not_guarantee_descent)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度下降"]},"variant":{"family_id":"qf-mod50-gd-one-step","seed":202611193036,"parameters":{"diagonal":[2,4],"linear":[-1,1],"point":[-2,0],"step_size":[1,4],"domain":"Exact_diagonal_quadratic_gradient_descent","operation":"gd-one-step","semantic_template":"gd-one-step|{\"diagonal\":[[2,1],[4,1]],\"linear\":[[-1,1],[1,1]],\"point\":[[-2,1],[0,1]],\"step_size\":[1,4]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r03-0037","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-descent"],"title_zh":"梯度下降","stem_zh":"standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-one-step；明示資料={\"diagonal\":[3,5],\"linear\":[0,2],\"point\":[-1,1],\"step_size\":[1,4]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([-1, 1], [-3, 7], 1/4, [-1/4, -3/4], 6, 0, 是, positive_step_alone_does_not_guarantee_descent)","explanation_zh":"Wave6第十三層核對=([-1, 1], [-3, 7], 1/4, [-1/4, -3/4], 6, 0, 是, positive_step_alone_does_not_guarantee_descent)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度下降"]},"variant":{"family_id":"qf-mod50-gd-one-step","seed":202611193037,"parameters":{"diagonal":[3,5],"linear":[0,2],"point":[-1,1],"step_size":[1,4],"domain":"Exact_diagonal_quadratic_gradient_descent","operation":"gd-one-step","semantic_template":"gd-one-step|{\"diagonal\":[[3,1],[5,1]],\"linear\":[[0,1],[2,1]],\"point\":[[-1,1],[1,1]],\"step_size\":[1,4]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r03-0038","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-descent"],"title_zh":"梯度下降","stem_zh":"standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-one-step；明示資料={\"diagonal\":[1,2],\"linear\":[1,3],\"point\":[0,2],\"step_size\":[1,4]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([0, 2], [1, 7], 1/4, [-1/4, 1/4], 10, 19/32, 是, positive_step_alone_does_not_guarantee_descent)","explanation_zh":"Wave6第十三層核對=([0, 2], [1, 7], 1/4, [-1/4, 1/4], 10, 19/32, 是, positive_step_alone_does_not_guarantee_descent)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度下降"]},"variant":{"family_id":"qf-mod50-gd-one-step","seed":202611193038,"parameters":{"diagonal":[1,2],"linear":[1,3],"point":[0,2],"step_size":[1,4],"domain":"Exact_diagonal_quadratic_gradient_descent","operation":"gd-one-step","semantic_template":"gd-one-step|{\"diagonal\":[[1,1],[2,1]],\"linear\":[[1,1],[3,1]],\"point\":[[0,1],[2,1]],\"step_size\":[1,4]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r03-0039","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-descent"],"title_zh":"梯度下降","stem_zh":"standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-one-step；明示資料={\"diagonal\":[2,3],\"linear\":[2,-3],\"point\":[1,3],\"step_size\":[1,4]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([1, 3], [4, 6], 1/4, [0, 3/2], 15/2, -9/8, 是, positive_step_alone_does_not_guarantee_descent)","explanation_zh":"Wave6第十三層核對=([1, 3], [4, 6], 1/4, [0, 3/2], 15/2, -9/8, 是, positive_step_alone_does_not_guarantee_descent)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度下降"]},"variant":{"family_id":"qf-mod50-gd-one-step","seed":202611193039,"parameters":{"diagonal":[2,3],"linear":[2,-3],"point":[1,3],"step_size":[1,4],"domain":"Exact_diagonal_quadratic_gradient_descent","operation":"gd-one-step","semantic_template":"gd-one-step|{\"diagonal\":[[2,1],[3,1]],\"linear\":[[2,1],[-3,1]],\"point\":[[1,1],[3,1]],\"step_size\":[1,4]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r03-0040","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-descent"],"title_zh":"梯度下降","stem_zh":"standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-one-step；明示資料={\"diagonal\":[3,4],\"linear\":[-2,-2],\"point\":[2,-3],\"step_size\":[1,4]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([2, -3], [4, -14], 1/4, [1, 1/2], 26, -1, 是, positive_step_alone_does_not_guarantee_descent)","explanation_zh":"Wave6第十三層核對=([2, -3], [4, -14], 1/4, [1, 1/2], 26, -1, 是, positive_step_alone_does_not_guarantee_descent)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度下降"]},"variant":{"family_id":"qf-mod50-gd-one-step","seed":202611193040,"parameters":{"diagonal":[3,4],"linear":[-2,-2],"point":[2,-3],"step_size":[1,4],"domain":"Exact_diagonal_quadratic_gradient_descent","operation":"gd-one-step","semantic_template":"gd-one-step|{\"diagonal\":[[3,1],[4,1]],\"linear\":[[-2,1],[-2,1]],\"point\":[[2,1],[-3,1]],\"step_size\":[1,4]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r03-0041","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-descent"],"title_zh":"梯度下降","stem_zh":"standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-one-step；明示資料={\"diagonal\":[1,5],\"linear\":[-1,-1],\"point\":[-2,-2],\"step_size\":[1,4]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([-2, -2], [-3, -11], 1/4, [-5/4, 3/4], 16, 43/16, 是, positive_step_alone_does_not_guarantee_descent)","explanation_zh":"Wave6第十三層核對=([-2, -2], [-3, -11], 1/4, [-5/4, 3/4], 16, 43/16, 是, positive_step_alone_does_not_guarantee_descent)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度下降"]},"variant":{"family_id":"qf-mod50-gd-one-step","seed":202611193041,"parameters":{"diagonal":[1,5],"linear":[-1,-1],"point":[-2,-2],"step_size":[1,4],"domain":"Exact_diagonal_quadratic_gradient_descent","operation":"gd-one-step","semantic_template":"gd-one-step|{\"diagonal\":[[1,1],[5,1]],\"linear\":[[-1,1],[-1,1]],\"point\":[[-2,1],[-2,1]],\"step_size\":[1,4]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r03-0042","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-descent"],"title_zh":"梯度下降","stem_zh":"standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-one-step；明示資料={\"diagonal\":[2,2],\"linear\":[0,0],\"point\":[-1,-1],\"step_size\":[1,4]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([-1, -1], [-2, -2], 1/4, [-1/2, -1/2], 2, 1/2, 是, positive_step_alone_does_not_guarantee_descent)","explanation_zh":"Wave6第十三層核對=([-1, -1], [-2, -2], 1/4, [-1/2, -1/2], 2, 1/2, 是, positive_step_alone_does_not_guarantee_descent)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度下降"]},"variant":{"family_id":"qf-mod50-gd-one-step","seed":202611193042,"parameters":{"diagonal":[2,2],"linear":[0,0],"point":[-1,-1],"step_size":[1,4],"domain":"Exact_diagonal_quadratic_gradient_descent","operation":"gd-one-step","semantic_template":"gd-one-step|{\"diagonal\":[[2,1],[2,1]],\"linear\":[[0,1],[0,1]],\"point\":[[-1,1],[-1,1]],\"step_size\":[1,4]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r03-0043","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-descent"],"title_zh":"梯度下降","stem_zh":"standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-one-step；明示資料={\"diagonal\":[3,3],\"linear\":[1,1],\"point\":[0,0],\"step_size\":[1,4]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([0, 0], [1, 1], 1/4, [-1/4, -1/4], 0, -5/16, 是, positive_step_alone_does_not_guarantee_descent)","explanation_zh":"Wave6第十三層核對=([0, 0], [1, 1], 1/4, [-1/4, -1/4], 0, -5/16, 是, positive_step_alone_does_not_guarantee_descent)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["梯度下降"]},"variant":{"family_id":"qf-mod50-gd-one-step","seed":202611193043,"parameters":{"diagonal":[3,3],"linear":[1,1],"point":[0,0],"step_size":[1,4],"domain":"Exact_diagonal_quadratic_gradient_descent","operation":"gd-one-step","semantic_template":"gd-one-step|{\"diagonal\":[[3,1],[3,1]],\"linear\":[[1,1],[1,1]],\"point\":[[0,1],[0,1]],\"step_size\":[1,4]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r03-0044","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-descent"],"title_zh":"梯度下降","stem_zh":"standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-one-step；明示資料={\"diagonal\":[1,4],\"linear\":[2,2],\"point\":[1,1],\"step_size\":[1,4]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([1, 1], [3, 6], 1/4, [1/4, -1/2], 13/2, 1/32, 是, positive_step_alone_does_not_guarantee_descent)","explanation_zh":"Wave6第十三層核對=([1, 1], [3, 6], 1/4, [1/4, -1/2], 13/2, 1/32, 是, positive_step_alone_does_not_guarantee_descent)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["梯度下降"]},"variant":{"family_id":"qf-mod50-gd-one-step","seed":202611193044,"parameters":{"diagonal":[1,4],"linear":[2,2],"point":[1,1],"step_size":[1,4],"domain":"Exact_diagonal_quadratic_gradient_descent","operation":"gd-one-step","semantic_template":"gd-one-step|{\"diagonal\":[[1,1],[4,1]],\"linear\":[[2,1],[2,1]],\"point\":[[1,1],[1,1]],\"step_size\":[1,4]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r03-0045","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-descent"],"title_zh":"梯度下降","stem_zh":"standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-one-step；明示資料={\"diagonal\":[2,5],\"linear\":[-2,3],\"point\":[2,2],\"step_size\":[1,4]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([2, 2], [2, 13], 1/4, [3/2, -5/4], 16, -19/32, 是, positive_step_alone_does_not_guarantee_descent)","explanation_zh":"Wave6第十三層核對=([2, 2], [2, 13], 1/4, [3/2, -5/4], 16, -19/32, 是, positive_step_alone_does_not_guarantee_descent)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["梯度下降"]},"variant":{"family_id":"qf-mod50-gd-one-step","seed":202611193045,"parameters":{"diagonal":[2,5],"linear":[-2,3],"point":[2,2],"step_size":[1,4],"domain":"Exact_diagonal_quadratic_gradient_descent","operation":"gd-one-step","semantic_template":"gd-one-step|{\"diagonal\":[[2,1],[5,1]],\"linear\":[[-2,1],[3,1]],\"point\":[[2,1],[2,1]],\"step_size\":[1,4]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r03-0046","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-descent"],"title_zh":"梯度下降","stem_zh":"standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-one-step；明示資料={\"diagonal\":[3,2],\"linear\":[-1,-3],\"point\":[-2,3],\"step_size\":[1,4]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([-2, 3], [-7, 3], 1/4, [-1/4, 9/4], 8, -43/32, 是, positive_step_alone_does_not_guarantee_descent)","explanation_zh":"Wave6第十三層核對=([-2, 3], [-7, 3], 1/4, [-1/4, 9/4], 8, -43/32, 是, positive_step_alone_does_not_guarantee_descent)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["梯度下降"]},"variant":{"family_id":"qf-mod50-gd-one-step","seed":202611193046,"parameters":{"diagonal":[3,2],"linear":[-1,-3],"point":[-2,3],"step_size":[1,4],"domain":"Exact_diagonal_quadratic_gradient_descent","operation":"gd-one-step","semantic_template":"gd-one-step|{\"diagonal\":[[3,1],[2,1]],\"linear\":[[-1,1],[-3,1]],\"point\":[[-2,1],[3,1]],\"step_size\":[1,4]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r03-0047","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-descent"],"title_zh":"梯度下降","stem_zh":"standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-one-step；明示資料={\"diagonal\":[1,3],\"linear\":[0,-2],\"point\":[-1,-3],\"step_size\":[1,4]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([-1, -3], [-1, -11], 1/4, [-3/4, -1/4], 20, 7/8, 是, positive_step_alone_does_not_guarantee_descent)","explanation_zh":"Wave6第十三層核對=([-1, -3], [-1, -11], 1/4, [-3/4, -1/4], 20, 7/8, 是, positive_step_alone_does_not_guarantee_descent)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["梯度下降"]},"variant":{"family_id":"qf-mod50-gd-one-step","seed":202611193047,"parameters":{"diagonal":[1,3],"linear":[0,-2],"point":[-1,-3],"step_size":[1,4],"domain":"Exact_diagonal_quadratic_gradient_descent","operation":"gd-one-step","semantic_template":"gd-one-step|{\"diagonal\":[[1,1],[3,1]],\"linear\":[[0,1],[-2,1]],\"point\":[[-1,1],[-3,1]],\"step_size\":[1,4]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r03-0048","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-descent"],"title_zh":"梯度下降","stem_zh":"standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-one-step；明示資料={\"diagonal\":[2,4],\"linear\":[1,-1],\"point\":[0,-2],\"step_size\":[1,4]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([0, -2], [1, -9], 1/4, [-1/4, 1/4], 10, -5/16, 是, positive_step_alone_does_not_guarantee_descent)","explanation_zh":"Wave6第十三層核對=([0, -2], [1, -9], 1/4, [-1/4, 1/4], 10, -5/16, 是, positive_step_alone_does_not_guarantee_descent)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["梯度下降"]},"variant":{"family_id":"qf-mod50-gd-one-step","seed":202611193048,"parameters":{"diagonal":[2,4],"linear":[1,-1],"point":[0,-2],"step_size":[1,4],"domain":"Exact_diagonal_quadratic_gradient_descent","operation":"gd-one-step","semantic_template":"gd-one-step|{\"diagonal\":[[2,1],[4,1]],\"linear\":[[1,1],[-1,1]],\"point\":[[0,1],[-2,1]],\"step_size\":[1,4]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r03-0049","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-descent"],"title_zh":"梯度下降","stem_zh":"standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-one-step；明示資料={\"diagonal\":[3,5],\"linear\":[2,0],\"point\":[1,-1],\"step_size\":[1,4]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([1, -1], [5, -5], 1/4, [-1/4, 1/4], 6, -1/4, 是, positive_step_alone_does_not_guarantee_descent)","explanation_zh":"Wave6第十三層核對=([1, -1], [5, -5], 1/4, [-1/4, 1/4], 6, -1/4, 是, positive_step_alone_does_not_guarantee_descent)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["梯度下降"]},"variant":{"family_id":"qf-mod50-gd-one-step","seed":202611193049,"parameters":{"diagonal":[3,5],"linear":[2,0],"point":[1,-1],"step_size":[1,4],"domain":"Exact_diagonal_quadratic_gradient_descent","operation":"gd-one-step","semantic_template":"gd-one-step|{\"diagonal\":[[3,1],[5,1]],\"linear\":[[2,1],[0,1]],\"point\":[[1,1],[-1,1]],\"step_size\":[1,4]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r03-0050","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-descent"],"title_zh":"梯度下降","stem_zh":"standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-one-step；明示資料={\"diagonal\":[1,2],\"linear\":[-2,1],\"point\":[2,0],\"step_size\":[1,4]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([2, 0], [0, 1], 1/4, [2, -1/4], -2, -35/16, 是, positive_step_alone_does_not_guarantee_descent)","explanation_zh":"Wave6第十三層核對=([2, 0], [0, 1], 1/4, [2, -1/4], -2, -35/16, 是, positive_step_alone_does_not_guarantee_descent)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["梯度下降"]},"variant":{"family_id":"qf-mod50-gd-one-step","seed":202611193050,"parameters":{"diagonal":[1,2],"linear":[-2,1],"point":[2,0],"step_size":[1,4],"domain":"Exact_diagonal_quadratic_gradient_descent","operation":"gd-one-step","semantic_template":"gd-one-step|{\"diagonal\":[[1,1],[2,1]],\"linear\":[[-2,1],[1,1]],\"point\":[[2,1],[0,1]],\"step_size\":[1,4]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r03-0051","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-descent"],"title_zh":"梯度下降","stem_zh":"standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-safe-quadratic-trace；明示資料={\"diagonal\":[1,2],\"linear\":[-1,-2],\"point\":[-2,-3],\"step_size\":[1,4],\"steps\":1}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(2, 1/4, 1, 1, [[0, [-2, -3], 19, [-3, -8]], [1, [-5/4, -1], 161/32, [-9/4, -4]]], 是, 否, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)","explanation_zh":"Wave6第十三層核對=(2, 1/4, 1, 1, [[0, [-2, -3], 19, [-3, -8]], [1, [-5/4, -1], 161/32, [-9/4, -4]]], 是, 否, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["梯度下降"]},"variant":{"family_id":"qf-mod50-gd-safe-quadratic-trace","seed":202611193051,"parameters":{"diagonal":[1,2],"linear":[-1,-2],"point":[-2,-3],"step_size":[1,4],"steps":1,"domain":"Exact_diagonal_quadratic_gradient_descent","operation":"gd-safe-quadratic-trace","semantic_template":"gd-safe-quadratic-trace|{\"diagonal\":[[1,1],[2,1]],\"linear\":[[-1,1],[-2,1]],\"point\":[[-2,1],[-3,1]],\"step_size\":[1,4],\"steps\":1}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r03-0052","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-descent"],"title_zh":"梯度下降","stem_zh":"standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-safe-quadratic-trace；明示資料={\"diagonal\":[2,3],\"linear\":[0,-1],\"point\":[-1,-2],\"step_size\":[1,4],\"steps\":1}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(3, 1/4, 2/3, 1, [[0, [-1, -2], 9, [-2, -7]], [1, [-1/2, -1/4], 19/32, [-1, -7/4]]], 是, 否, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)","explanation_zh":"Wave6第十三層核對=(3, 1/4, 2/3, 1, [[0, [-1, -2], 9, [-2, -7]], [1, [-1/2, -1/4], 19/32, [-1, -7/4]]], 是, 否, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["梯度下降"]},"variant":{"family_id":"qf-mod50-gd-safe-quadratic-trace","seed":202611193052,"parameters":{"diagonal":[2,3],"linear":[0,-1],"point":[-1,-2],"step_size":[1,4],"steps":1,"domain":"Exact_diagonal_quadratic_gradient_descent","operation":"gd-safe-quadratic-trace","semantic_template":"gd-safe-quadratic-trace|{\"diagonal\":[[2,1],[3,1]],\"linear\":[[0,1],[-1,1]],\"point\":[[-1,1],[-2,1]],\"step_size\":[1,4],\"steps\":1}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r03-0053","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-descent"],"title_zh":"梯度下降","stem_zh":"standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-safe-quadratic-trace；明示資料={\"diagonal\":[3,4],\"linear\":[1,0],\"point\":[0,-1],\"step_size\":[1,4],\"steps\":1}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(4, 1/4, 1/2, 1, [[0, [0, -1], 2, [1, -4]], [1, [-1/4, 0], -5/32, [1/4, 0]]], 是, 否, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)","explanation_zh":"Wave6第十三層核對=(4, 1/4, 1/2, 1, [[0, [0, -1], 2, [1, -4]], [1, [-1/4, 0], -5/32, [1/4, 0]]], 是, 否, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["梯度下降"]},"variant":{"family_id":"qf-mod50-gd-safe-quadratic-trace","seed":202611193053,"parameters":{"diagonal":[3,4],"linear":[1,0],"point":[0,-1],"step_size":[1,4],"steps":1,"domain":"Exact_diagonal_quadratic_gradient_descent","operation":"gd-safe-quadratic-trace","semantic_template":"gd-safe-quadratic-trace|{\"diagonal\":[[3,1],[4,1]],\"linear\":[[1,1],[0,1]],\"point\":[[0,1],[-1,1]],\"step_size\":[1,4],\"steps\":1}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r03-0054","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-descent"],"title_zh":"梯度下降","stem_zh":"standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-safe-quadratic-trace；明示資料={\"diagonal\":[1,5],\"linear\":[2,1],\"point\":[1,0],\"step_size\":[1,4],\"steps\":1}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(5, 1/4, 2/5, 1, [[0, [1, 0], 5/2, [3, 1]], [1, [1/4, -1/4], 7/16, [9/4, -1/4]]], 是, 否, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)","explanation_zh":"Wave6第十三層核對=(5, 1/4, 2/5, 1, [[0, [1, 0], 5/2, [3, 1]], [1, [1/4, -1/4], 7/16, [9/4, -1/4]]], 是, 否, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["梯度下降"]},"variant":{"family_id":"qf-mod50-gd-safe-quadratic-trace","seed":202611193054,"parameters":{"diagonal":[1,5],"linear":[2,1],"point":[1,0],"step_size":[1,4],"steps":1,"domain":"Exact_diagonal_quadratic_gradient_descent","operation":"gd-safe-quadratic-trace","semantic_template":"gd-safe-quadratic-trace|{\"diagonal\":[[1,1],[5,1]],\"linear\":[[2,1],[1,1]],\"point\":[[1,1],[0,1]],\"step_size\":[1,4],\"steps\":1}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r03-0055","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-descent"],"title_zh":"梯度下降","stem_zh":"standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-safe-quadratic-trace；明示資料={\"diagonal\":[2,2],\"linear\":[-2,2],\"point\":[2,1],\"step_size\":[1,4],\"steps\":1}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(2, 1/4, 1, 1, [[0, [2, 1], 3, [2, 4]], [1, [3/2, 0], -3/4, [1, 2]]], 是, 否, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)","explanation_zh":"Wave6第十三層核對=(2, 1/4, 1, 1, [[0, [2, 1], 3, [2, 4]], [1, [3/2, 0], -3/4, [1, 2]]], 是, 否, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["梯度下降"]},"variant":{"family_id":"qf-mod50-gd-safe-quadratic-trace","seed":202611193055,"parameters":{"diagonal":[2,2],"linear":[-2,2],"point":[2,1],"step_size":[1,4],"steps":1,"domain":"Exact_diagonal_quadratic_gradient_descent","operation":"gd-safe-quadratic-trace","semantic_template":"gd-safe-quadratic-trace|{\"diagonal\":[[2,1],[2,1]],\"linear\":[[-2,1],[2,1]],\"point\":[[2,1],[1,1]],\"step_size\":[1,4],\"steps\":1}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r03-0056","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-descent"],"title_zh":"梯度下降","stem_zh":"standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-safe-quadratic-trace；明示資料={\"diagonal\":[3,3],\"linear\":[-1,3],\"point\":[-2,2],\"step_size\":[1,4],\"steps\":1}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(3, 1/4, 2/3, 1, [[0, [-2, 2], 20, [-7, 9]], [1, [-1/4, -1/4], -5/16, [-7/4, 9/4]]], 是, 否, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)","explanation_zh":"Wave6第十三層核對=(3, 1/4, 2/3, 1, [[0, [-2, 2], 20, [-7, 9]], [1, [-1/4, -1/4], -5/16, [-7/4, 9/4]]], 是, 否, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["梯度下降"]},"variant":{"family_id":"qf-mod50-gd-safe-quadratic-trace","seed":202611193056,"parameters":{"diagonal":[3,3],"linear":[-1,3],"point":[-2,2],"step_size":[1,4],"steps":1,"domain":"Exact_diagonal_quadratic_gradient_descent","operation":"gd-safe-quadratic-trace","semantic_template":"gd-safe-quadratic-trace|{\"diagonal\":[[3,1],[3,1]],\"linear\":[[-1,1],[3,1]],\"point\":[[-2,1],[2,1]],\"step_size\":[1,4],\"steps\":1}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r03-0057","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-descent"],"title_zh":"梯度下降","stem_zh":"standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-safe-quadratic-trace；明示資料={\"diagonal\":[1,4],\"linear\":[0,-3],\"point\":[-1,3],\"step_size\":[1,4],\"steps\":1}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(4, 1/4, 1/2, 1, [[0, [-1, 3], 19/2, [-1, 9]], [1, [-3/4, 3/4], -27/32, [-3/4, 0]]], 是, 否, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)","explanation_zh":"Wave6第十三層核對=(4, 1/4, 1/2, 1, [[0, [-1, 3], 19/2, [-1, 9]], [1, [-3/4, 3/4], -27/32, [-3/4, 0]]], 是, 否, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["梯度下降"]},"variant":{"family_id":"qf-mod50-gd-safe-quadratic-trace","seed":202611193057,"parameters":{"diagonal":[1,4],"linear":[0,-3],"point":[-1,3],"step_size":[1,4],"steps":1,"domain":"Exact_diagonal_quadratic_gradient_descent","operation":"gd-safe-quadratic-trace","semantic_template":"gd-safe-quadratic-trace|{\"diagonal\":[[1,1],[4,1]],\"linear\":[[0,1],[-3,1]],\"point\":[[-1,1],[3,1]],\"step_size\":[1,4],\"steps\":1}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r03-0058","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-descent"],"title_zh":"梯度下降","stem_zh":"standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-safe-quadratic-trace；明示資料={\"diagonal\":[2,5],\"linear\":[1,-2],\"point\":[0,-3],\"step_size\":[1,4],\"steps\":1}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(5, 1/4, 2/5, 1, [[0, [0, -3], 57/2, [1, -17]], [1, [-1/4, 5/4], 39/32, [1/2, 17/4]]], 是, 否, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)","explanation_zh":"Wave6第十三層核對=(5, 1/4, 2/5, 1, [[0, [0, -3], 57/2, [1, -17]], [1, [-1/4, 5/4], 39/32, [1/2, 17/4]]], 是, 否, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["梯度下降"]},"variant":{"family_id":"qf-mod50-gd-safe-quadratic-trace","seed":202611193058,"parameters":{"diagonal":[2,5],"linear":[1,-2],"point":[0,-3],"step_size":[1,4],"steps":1,"domain":"Exact_diagonal_quadratic_gradient_descent","operation":"gd-safe-quadratic-trace","semantic_template":"gd-safe-quadratic-trace|{\"diagonal\":[[2,1],[5,1]],\"linear\":[[1,1],[-2,1]],\"point\":[[0,1],[-3,1]],\"step_size\":[1,4],\"steps\":1}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r03-0059","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-descent"],"title_zh":"梯度下降","stem_zh":"standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-safe-quadratic-trace；明示資料={\"diagonal\":[3,2],\"linear\":[2,-1],\"point\":[1,-2],\"step_size\":[1,4],\"steps\":1}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(3, 1/4, 2/3, 1, [[0, [1, -2], 19/2, [5, -5]], [1, [-1/4, -3/4], 29/32, [5/4, -5/2]]], 是, 否, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)","explanation_zh":"Wave6第十三層核對=(3, 1/4, 2/3, 1, [[0, [1, -2], 19/2, [5, -5]], [1, [-1/4, -3/4], 29/32, [5/4, -5/2]]], 是, 否, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度下降"]},"variant":{"family_id":"qf-mod50-gd-safe-quadratic-trace","seed":202611193059,"parameters":{"diagonal":[3,2],"linear":[2,-1],"point":[1,-2],"step_size":[1,4],"steps":1,"domain":"Exact_diagonal_quadratic_gradient_descent","operation":"gd-safe-quadratic-trace","semantic_template":"gd-safe-quadratic-trace|{\"diagonal\":[[3,1],[2,1]],\"linear\":[[2,1],[-1,1]],\"point\":[[1,1],[-2,1]],\"step_size\":[1,4],\"steps\":1}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r03-0060","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-descent"],"title_zh":"梯度下降","stem_zh":"standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-safe-quadratic-trace；明示資料={\"diagonal\":[1,3],\"linear\":[-2,0],\"point\":[2,-1],\"step_size\":[1,4],\"steps\":1}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(3, 1/4, 2/3, 1, [[0, [2, -1], -1/2, [0, -3]], [1, [2, -1/4], -61/32, [0, -3/4]]], 是, 否, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)","explanation_zh":"Wave6第十三層核對=(3, 1/4, 2/3, 1, [[0, [2, -1], -1/2, [0, -3]], [1, [2, -1/4], -61/32, [0, -3/4]]], 是, 否, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度下降"]},"variant":{"family_id":"qf-mod50-gd-safe-quadratic-trace","seed":202611193060,"parameters":{"diagonal":[1,3],"linear":[-2,0],"point":[2,-1],"step_size":[1,4],"steps":1,"domain":"Exact_diagonal_quadratic_gradient_descent","operation":"gd-safe-quadratic-trace","semantic_template":"gd-safe-quadratic-trace|{\"diagonal\":[[1,1],[3,1]],\"linear\":[[-2,1],[0,1]],\"point\":[[2,1],[-1,1]],\"step_size\":[1,4],\"steps\":1}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r03-0061","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-descent"],"title_zh":"梯度下降","stem_zh":"standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-safe-quadratic-trace；明示資料={\"diagonal\":[2,4],\"linear\":[-1,1],\"point\":[-2,0],\"step_size\":[1,4],\"steps\":1}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(4, 1/4, 1/2, 1, [[0, [-2, 0], 6, [-5, 1]], [1, [-3/4, -1/4], 19/16, [-5/2, 0]]], 是, 否, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)","explanation_zh":"Wave6第十三層核對=(4, 1/4, 1/2, 1, [[0, [-2, 0], 6, [-5, 1]], [1, [-3/4, -1/4], 19/16, [-5/2, 0]]], 是, 否, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度下降"]},"variant":{"family_id":"qf-mod50-gd-safe-quadratic-trace","seed":202611193061,"parameters":{"diagonal":[2,4],"linear":[-1,1],"point":[-2,0],"step_size":[1,4],"steps":1,"domain":"Exact_diagonal_quadratic_gradient_descent","operation":"gd-safe-quadratic-trace","semantic_template":"gd-safe-quadratic-trace|{\"diagonal\":[[2,1],[4,1]],\"linear\":[[-1,1],[1,1]],\"point\":[[-2,1],[0,1]],\"step_size\":[1,4],\"steps\":1}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r03-0062","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-descent"],"title_zh":"梯度下降","stem_zh":"standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-safe-quadratic-trace；明示資料={\"diagonal\":[3,5],\"linear\":[0,2],\"point\":[-1,1],\"step_size\":[1,4],\"steps\":1}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(5, 1/4, 2/5, 1, [[0, [-1, 1], 6, [-3, 7]], [1, [-1/4, -3/4], 0, [-3/4, -7/4]]], 是, 否, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)","explanation_zh":"Wave6第十三層核對=(5, 1/4, 2/5, 1, [[0, [-1, 1], 6, [-3, 7]], [1, [-1/4, -3/4], 0, [-3/4, -7/4]]], 是, 否, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度下降"]},"variant":{"family_id":"qf-mod50-gd-safe-quadratic-trace","seed":202611193062,"parameters":{"diagonal":[3,5],"linear":[0,2],"point":[-1,1],"step_size":[1,4],"steps":1,"domain":"Exact_diagonal_quadratic_gradient_descent","operation":"gd-safe-quadratic-trace","semantic_template":"gd-safe-quadratic-trace|{\"diagonal\":[[3,1],[5,1]],\"linear\":[[0,1],[2,1]],\"point\":[[-1,1],[1,1]],\"step_size\":[1,4],\"steps\":1}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r03-0063","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-descent"],"title_zh":"梯度下降","stem_zh":"standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-safe-quadratic-trace；明示資料={\"diagonal\":[1,2],\"linear\":[1,3],\"point\":[0,2],\"step_size\":[1,4],\"steps\":1}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(2, 1/4, 1, 1, [[0, [0, 2], 10, [1, 7]], [1, [-1/4, 1/4], 19/32, [3/4, 7/2]]], 是, 否, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)","explanation_zh":"Wave6第十三層核對=(2, 1/4, 1, 1, [[0, [0, 2], 10, [1, 7]], [1, [-1/4, 1/4], 19/32, [3/4, 7/2]]], 是, 否, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度下降"]},"variant":{"family_id":"qf-mod50-gd-safe-quadratic-trace","seed":202611193063,"parameters":{"diagonal":[1,2],"linear":[1,3],"point":[0,2],"step_size":[1,4],"steps":1,"domain":"Exact_diagonal_quadratic_gradient_descent","operation":"gd-safe-quadratic-trace","semantic_template":"gd-safe-quadratic-trace|{\"diagonal\":[[1,1],[2,1]],\"linear\":[[1,1],[3,1]],\"point\":[[0,1],[2,1]],\"step_size\":[1,4],\"steps\":1}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r03-0064","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-descent"],"title_zh":"梯度下降","stem_zh":"standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-safe-quadratic-trace；明示資料={\"diagonal\":[2,3],\"linear\":[2,-3],\"point\":[1,3],\"step_size\":[1,4],\"steps\":1}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(3, 1/4, 2/3, 1, [[0, [1, 3], 15/2, [4, 6]], [1, [0, 3/2], -9/8, [2, 3/2]]], 是, 否, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)","explanation_zh":"Wave6第十三層核對=(3, 1/4, 2/3, 1, [[0, [1, 3], 15/2, [4, 6]], [1, [0, 3/2], -9/8, [2, 3/2]]], 是, 否, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度下降"]},"variant":{"family_id":"qf-mod50-gd-safe-quadratic-trace","seed":202611193064,"parameters":{"diagonal":[2,3],"linear":[2,-3],"point":[1,3],"step_size":[1,4],"steps":1,"domain":"Exact_diagonal_quadratic_gradient_descent","operation":"gd-safe-quadratic-trace","semantic_template":"gd-safe-quadratic-trace|{\"diagonal\":[[2,1],[3,1]],\"linear\":[[2,1],[-3,1]],\"point\":[[1,1],[3,1]],\"step_size\":[1,4],\"steps\":1}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r03-0065","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-descent"],"title_zh":"梯度下降","stem_zh":"standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-safe-quadratic-trace；明示資料={\"diagonal\":[3,4],\"linear\":[-2,-2],\"point\":[2,-3],\"step_size\":[1,4],\"steps\":1}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(4, 1/4, 1/2, 1, [[0, [2, -3], 26, [4, -14]], [1, [1, 1/2], -1, [1, 0]]], 是, 否, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)","explanation_zh":"Wave6第十三層核對=(4, 1/4, 1/2, 1, [[0, [2, -3], 26, [4, -14]], [1, [1, 1/2], -1, [1, 0]]], 是, 否, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度下降"]},"variant":{"family_id":"qf-mod50-gd-safe-quadratic-trace","seed":202611193065,"parameters":{"diagonal":[3,4],"linear":[-2,-2],"point":[2,-3],"step_size":[1,4],"steps":1,"domain":"Exact_diagonal_quadratic_gradient_descent","operation":"gd-safe-quadratic-trace","semantic_template":"gd-safe-quadratic-trace|{\"diagonal\":[[3,1],[4,1]],\"linear\":[[-2,1],[-2,1]],\"point\":[[2,1],[-3,1]],\"step_size\":[1,4],\"steps\":1}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r03-0066","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-descent"],"title_zh":"梯度下降","stem_zh":"standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-safe-quadratic-trace；明示資料={\"diagonal\":[1,5],\"linear\":[-1,-1],\"point\":[-2,-2],\"step_size\":[1,4],\"steps\":1}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(5, 1/4, 2/5, 1, [[0, [-2, -2], 16, [-3, -11]], [1, [-5/4, 3/4], 43/16, [-9/4, 11/4]]], 是, 否, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)","explanation_zh":"Wave6第十三層核對=(5, 1/4, 2/5, 1, [[0, [-2, -2], 16, [-3, -11]], [1, [-5/4, 3/4], 43/16, [-9/4, 11/4]]], 是, 否, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度下降"]},"variant":{"family_id":"qf-mod50-gd-safe-quadratic-trace","seed":202611193066,"parameters":{"diagonal":[1,5],"linear":[-1,-1],"point":[-2,-2],"step_size":[1,4],"steps":1,"domain":"Exact_diagonal_quadratic_gradient_descent","operation":"gd-safe-quadratic-trace","semantic_template":"gd-safe-quadratic-trace|{\"diagonal\":[[1,1],[5,1]],\"linear\":[[-1,1],[-1,1]],\"point\":[[-2,1],[-2,1]],\"step_size\":[1,4],\"steps\":1}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r03-0067","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-descent"],"title_zh":"梯度下降","stem_zh":"standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-safe-quadratic-trace；明示資料={\"diagonal\":[2,2],\"linear\":[0,0],\"point\":[-1,-1],\"step_size\":[1,4],\"steps\":1}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(2, 1/4, 1, 1, [[0, [-1, -1], 2, [-2, -2]], [1, [-1/2, -1/2], 1/2, [-1, -1]]], 是, 否, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)","explanation_zh":"Wave6第十三層核對=(2, 1/4, 1, 1, [[0, [-1, -1], 2, [-2, -2]], [1, [-1/2, -1/2], 1/2, [-1, -1]]], 是, 否, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度下降"]},"variant":{"family_id":"qf-mod50-gd-safe-quadratic-trace","seed":202611193067,"parameters":{"diagonal":[2,2],"linear":[0,0],"point":[-1,-1],"step_size":[1,4],"steps":1,"domain":"Exact_diagonal_quadratic_gradient_descent","operation":"gd-safe-quadratic-trace","semantic_template":"gd-safe-quadratic-trace|{\"diagonal\":[[2,1],[2,1]],\"linear\":[[0,1],[0,1]],\"point\":[[-1,1],[-1,1]],\"step_size\":[1,4],\"steps\":1}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r03-0068","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-descent"],"title_zh":"梯度下降","stem_zh":"standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-safe-quadratic-trace；明示資料={\"diagonal\":[3,3],\"linear\":[1,1],\"point\":[0,0],\"step_size\":[1,4],\"steps\":1}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(3, 1/4, 2/3, 1, [[0, [0, 0], 0, [1, 1]], [1, [-1/4, -1/4], -5/16, [1/4, 1/4]]], 是, 否, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)","explanation_zh":"Wave6第十三層核對=(3, 1/4, 2/3, 1, [[0, [0, 0], 0, [1, 1]], [1, [-1/4, -1/4], -5/16, [1/4, 1/4]]], 是, 否, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["梯度下降"]},"variant":{"family_id":"qf-mod50-gd-safe-quadratic-trace","seed":202611193068,"parameters":{"diagonal":[3,3],"linear":[1,1],"point":[0,0],"step_size":[1,4],"steps":1,"domain":"Exact_diagonal_quadratic_gradient_descent","operation":"gd-safe-quadratic-trace","semantic_template":"gd-safe-quadratic-trace|{\"diagonal\":[[3,1],[3,1]],\"linear\":[[1,1],[1,1]],\"point\":[[0,1],[0,1]],\"step_size\":[1,4],\"steps\":1}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r03-0069","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-descent"],"title_zh":"梯度下降","stem_zh":"standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-safe-quadratic-trace；明示資料={\"diagonal\":[1,4],\"linear\":[2,2],\"point\":[1,1],\"step_size\":[1,4],\"steps\":1}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(4, 1/4, 1/2, 1, [[0, [1, 1], 13/2, [3, 6]], [1, [1/4, -1/2], 1/32, [9/4, 0]]], 是, 否, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)","explanation_zh":"Wave6第十三層核對=(4, 1/4, 1/2, 1, [[0, [1, 1], 13/2, [3, 6]], [1, [1/4, -1/2], 1/32, [9/4, 0]]], 是, 否, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["梯度下降"]},"variant":{"family_id":"qf-mod50-gd-safe-quadratic-trace","seed":202611193069,"parameters":{"diagonal":[1,4],"linear":[2,2],"point":[1,1],"step_size":[1,4],"steps":1,"domain":"Exact_diagonal_quadratic_gradient_descent","operation":"gd-safe-quadratic-trace","semantic_template":"gd-safe-quadratic-trace|{\"diagonal\":[[1,1],[4,1]],\"linear\":[[2,1],[2,1]],\"point\":[[1,1],[1,1]],\"step_size\":[1,4],\"steps\":1}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r03-0070","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-descent"],"title_zh":"梯度下降","stem_zh":"standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-safe-quadratic-trace；明示資料={\"diagonal\":[2,5],\"linear\":[-2,3],\"point\":[2,2],\"step_size\":[1,4],\"steps\":1}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(5, 1/4, 2/5, 1, [[0, [2, 2], 16, [2, 13]], [1, [3/2, -5/4], -19/32, [1, -13/4]]], 是, 否, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)","explanation_zh":"Wave6第十三層核對=(5, 1/4, 2/5, 1, [[0, [2, 2], 16, [2, 13]], [1, [3/2, -5/4], -19/32, [1, -13/4]]], 是, 否, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["梯度下降"]},"variant":{"family_id":"qf-mod50-gd-safe-quadratic-trace","seed":202611193070,"parameters":{"diagonal":[2,5],"linear":[-2,3],"point":[2,2],"step_size":[1,4],"steps":1,"domain":"Exact_diagonal_quadratic_gradient_descent","operation":"gd-safe-quadratic-trace","semantic_template":"gd-safe-quadratic-trace|{\"diagonal\":[[2,1],[5,1]],\"linear\":[[-2,1],[3,1]],\"point\":[[2,1],[2,1]],\"step_size\":[1,4],\"steps\":1}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r03-0071","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-descent"],"title_zh":"梯度下降","stem_zh":"standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-safe-quadratic-trace；明示資料={\"diagonal\":[3,2],\"linear\":[-1,-3],\"point\":[-2,3],\"step_size\":[1,4],\"steps\":1}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(3, 1/4, 2/3, 1, [[0, [-2, 3], 8, [-7, 3]], [1, [-1/4, 9/4], -43/32, [-7/4, 3/2]]], 是, 否, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)","explanation_zh":"Wave6第十三層核對=(3, 1/4, 2/3, 1, [[0, [-2, 3], 8, [-7, 3]], [1, [-1/4, 9/4], -43/32, [-7/4, 3/2]]], 是, 否, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["梯度下降"]},"variant":{"family_id":"qf-mod50-gd-safe-quadratic-trace","seed":202611193071,"parameters":{"diagonal":[3,2],"linear":[-1,-3],"point":[-2,3],"step_size":[1,4],"steps":1,"domain":"Exact_diagonal_quadratic_gradient_descent","operation":"gd-safe-quadratic-trace","semantic_template":"gd-safe-quadratic-trace|{\"diagonal\":[[3,1],[2,1]],\"linear\":[[-1,1],[-3,1]],\"point\":[[-2,1],[3,1]],\"step_size\":[1,4],\"steps\":1}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r03-0072","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-descent"],"title_zh":"梯度下降","stem_zh":"standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-safe-quadratic-trace；明示資料={\"diagonal\":[1,3],\"linear\":[0,-2],\"point\":[-1,-3],\"step_size\":[1,4],\"steps\":1}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(3, 1/4, 2/3, 1, [[0, [-1, -3], 20, [-1, -11]], [1, [-3/4, -1/4], 7/8, [-3/4, -11/4]]], 是, 否, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)","explanation_zh":"Wave6第十三層核對=(3, 1/4, 2/3, 1, [[0, [-1, -3], 20, [-1, -11]], [1, [-3/4, -1/4], 7/8, [-3/4, -11/4]]], 是, 否, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["梯度下降"]},"variant":{"family_id":"qf-mod50-gd-safe-quadratic-trace","seed":202611193072,"parameters":{"diagonal":[1,3],"linear":[0,-2],"point":[-1,-3],"step_size":[1,4],"steps":1,"domain":"Exact_diagonal_quadratic_gradient_descent","operation":"gd-safe-quadratic-trace","semantic_template":"gd-safe-quadratic-trace|{\"diagonal\":[[1,1],[3,1]],\"linear\":[[0,1],[-2,1]],\"point\":[[-1,1],[-3,1]],\"step_size\":[1,4],\"steps\":1}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r03-0073","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-descent"],"title_zh":"梯度下降","stem_zh":"standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-safe-quadratic-trace；明示資料={\"diagonal\":[2,4],\"linear\":[1,-1],\"point\":[0,-2],\"step_size\":[1,4],\"steps\":1}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(4, 1/4, 1/2, 1, [[0, [0, -2], 10, [1, -9]], [1, [-1/4, 1/4], -5/16, [1/2, 0]]], 是, 否, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)","explanation_zh":"Wave6第十三層核對=(4, 1/4, 1/2, 1, [[0, [0, -2], 10, [1, -9]], [1, [-1/4, 1/4], -5/16, [1/2, 0]]], 是, 否, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["梯度下降"]},"variant":{"family_id":"qf-mod50-gd-safe-quadratic-trace","seed":202611193073,"parameters":{"diagonal":[2,4],"linear":[1,-1],"point":[0,-2],"step_size":[1,4],"steps":1,"domain":"Exact_diagonal_quadratic_gradient_descent","operation":"gd-safe-quadratic-trace","semantic_template":"gd-safe-quadratic-trace|{\"diagonal\":[[2,1],[4,1]],\"linear\":[[1,1],[-1,1]],\"point\":[[0,1],[-2,1]],\"step_size\":[1,4],\"steps\":1}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r03-0074","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-descent"],"title_zh":"梯度下降","stem_zh":"standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-safe-quadratic-trace；明示資料={\"diagonal\":[3,5],\"linear\":[2,0],\"point\":[1,-1],\"step_size\":[1,4],\"steps\":1}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(5, 1/4, 2/5, 1, [[0, [1, -1], 6, [5, -5]], [1, [-1/4, 1/4], -1/4, [5/4, 5/4]]], 是, 否, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)","explanation_zh":"Wave6第十三層核對=(5, 1/4, 2/5, 1, [[0, [1, -1], 6, [5, -5]], [1, [-1/4, 1/4], -1/4, [5/4, 5/4]]], 是, 否, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["梯度下降"]},"variant":{"family_id":"qf-mod50-gd-safe-quadratic-trace","seed":202611193074,"parameters":{"diagonal":[3,5],"linear":[2,0],"point":[1,-1],"step_size":[1,4],"steps":1,"domain":"Exact_diagonal_quadratic_gradient_descent","operation":"gd-safe-quadratic-trace","semantic_template":"gd-safe-quadratic-trace|{\"diagonal\":[[3,1],[5,1]],\"linear\":[[2,1],[0,1]],\"point\":[[1,1],[-1,1]],\"step_size\":[1,4],\"steps\":1}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r03-0075","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-descent"],"title_zh":"梯度下降","stem_zh":"standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-safe-quadratic-trace；明示資料={\"diagonal\":[1,2],\"linear\":[-2,1],\"point\":[2,0],\"step_size\":[1,4],\"steps\":1}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(2, 1/4, 1, 1, [[0, [2, 0], -2, [0, 1]], [1, [2, -1/4], -35/16, [0, 1/2]]], 是, 否, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)","explanation_zh":"Wave6第十三層核對=(2, 1/4, 1, 1, [[0, [2, 0], -2, [0, 1]], [1, [2, -1/4], -35/16, [0, 1/2]]], 是, 否, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["梯度下降"]},"variant":{"family_id":"qf-mod50-gd-safe-quadratic-trace","seed":202611193075,"parameters":{"diagonal":[1,2],"linear":[-2,1],"point":[2,0],"step_size":[1,4],"steps":1,"domain":"Exact_diagonal_quadratic_gradient_descent","operation":"gd-safe-quadratic-trace","semantic_template":"gd-safe-quadratic-trace|{\"diagonal\":[[1,1],[2,1]],\"linear\":[[-2,1],[1,1]],\"point\":[[2,1],[0,1]],\"step_size\":[1,4],\"steps\":1}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r03-0076","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-descent"],"title_zh":"梯度下降","stem_zh":"standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-boundary；明示資料={\"mode\":\"oversized_step\",\"model\":{\"curvature\":1,\"point\":1,\"step_size\":3}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(oversized_step, 1, 1, 3, -2, 1/2, 2, 否, positive_but_oversized_step_can_increase_the_objective)","explanation_zh":"Wave6第十三層核對=(oversized_step, 1, 1, 3, -2, 1/2, 2, 否, positive_but_oversized_step_can_increase_the_objective)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["梯度下降"]},"variant":{"family_id":"qf-mod50-gd-boundary","seed":202611193076,"parameters":{"mode":"oversized_step","model":{"curvature":1,"point":1,"step_size":3},"domain":"Exact_diagonal_quadratic_gradient_descent","operation":"gd-boundary","semantic_template":"gd-boundary|{\"mode\":\"oversized_step\",\"model\":{\"curvature\":[1,1],\"point\":[1,1],\"step_size\":[3,1]}}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r03-0077","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-descent"],"title_zh":"梯度下降","stem_zh":"standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-boundary；明示資料={\"mode\":\"zero_gradient_nonminimum\",\"model\":{\"scale\":2}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(zero_gradient_nonminimum, f(x)=-2*x^2, x=0, gradient=0, strict_local_maximum, 否, zero_gradient_is_stationary_not_a_minimum_certificate)","explanation_zh":"Wave6第十三層核對=(zero_gradient_nonminimum, f(x)=-2*x^2, x=0, gradient=0, strict_local_maximum, 否, zero_gradient_is_stationary_not_a_minimum_certificate)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["梯度下降"]},"variant":{"family_id":"qf-mod50-gd-boundary","seed":202611193077,"parameters":{"mode":"zero_gradient_nonminimum","model":{"scale":2},"domain":"Exact_diagonal_quadratic_gradient_descent","operation":"gd-boundary","semantic_template":"gd-boundary|{\"mode\":\"zero_gradient_nonminimum\",\"model\":{\"scale\":[2,1]}}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r03-0078","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-descent"],"title_zh":"梯度下降","stem_zh":"standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-boundary；明示資料={\"mode\":\"finite_trace_only\",\"model\":{\"diagonal\":[1,2],\"linear\":[2,-2],\"point\":[3,1],\"step_size\":[1,4],\"steps\":1}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(finite_trace_only, 1, [[0, [3, 1], 19/2], [1, [7/4, 1], 129/32]], 否, finite_trace_does_not_prove_convergence_or_global_optimality)","explanation_zh":"Wave6第十三層核對=(finite_trace_only, 1, [[0, [3, 1], 19/2], [1, [7/4, 1], 129/32]], 否, finite_trace_does_not_prove_convergence_or_global_optimality)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["梯度下降"]},"variant":{"family_id":"qf-mod50-gd-boundary","seed":202611193078,"parameters":{"mode":"finite_trace_only","model":{"diagonal":[1,2],"linear":[2,-2],"point":[3,1],"step_size":[1,4],"steps":1},"domain":"Exact_diagonal_quadratic_gradient_descent","operation":"gd-boundary","semantic_template":"gd-boundary|{\"mode\":\"finite_trace_only\",\"model\":{\"diagonal\":[[1,1],[2,1]],\"linear\":[[2,1],[-2,1]],\"point\":[[3,1],[1,1]],\"step_size\":[1,4],\"steps\":1}}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r03-0079","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-descent"],"title_zh":"梯度下降","stem_zh":"standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-boundary；明示資料={\"mode\":\"constrained_confusion\",\"model\":{\"gradient\":4,\"lower\":-1,\"point\":0,\"step_size\":2,\"upper\":1}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(constrained_confusion, -1, 1, 0, -8, 否, unconstrained_gradient_step_is_not_projected_or_constrained_descent)","explanation_zh":"Wave6第十三層核對=(constrained_confusion, -1, 1, 0, -8, 否, unconstrained_gradient_step_is_not_projected_or_constrained_descent)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["梯度下降"]},"variant":{"family_id":"qf-mod50-gd-boundary","seed":202611193079,"parameters":{"mode":"constrained_confusion","model":{"point":0,"gradient":4,"step_size":2,"lower":-1,"upper":1},"domain":"Exact_diagonal_quadratic_gradient_descent","operation":"gd-boundary","semantic_template":"gd-boundary|{\"mode\":\"constrained_confusion\",\"model\":{\"gradient\":[4,1],\"lower\":[-1,1],\"point\":[0,1],\"step_size\":[2,1],\"upper\":[1,1]}}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r03-0080","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-descent"],"title_zh":"梯度下降","stem_zh":"standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-boundary；明示資料={\"mode\":\"oversized_step\",\"model\":{\"curvature\":5,\"point\":5,\"step_size\":3}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(oversized_step, 5, 5, 3, -70, 125/2, 12250, 否, positive_but_oversized_step_can_increase_the_objective)","explanation_zh":"Wave6第十三層核對=(oversized_step, 5, 5, 3, -70, 125/2, 12250, 否, positive_but_oversized_step_can_increase_the_objective)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["梯度下降"]},"variant":{"family_id":"qf-mod50-gd-boundary","seed":202611193080,"parameters":{"mode":"oversized_step","model":{"curvature":5,"point":5,"step_size":3},"domain":"Exact_diagonal_quadratic_gradient_descent","operation":"gd-boundary","semantic_template":"gd-boundary|{\"mode\":\"oversized_step\",\"model\":{\"curvature\":[5,1],\"point\":[5,1],\"step_size\":[3,1]}}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r03-0081","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-descent"],"title_zh":"梯度下降","stem_zh":"standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-boundary；明示資料={\"mode\":\"zero_gradient_nonminimum\",\"model\":{\"scale\":6}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(zero_gradient_nonminimum, f(x)=-6*x^2, x=0, gradient=0, strict_local_maximum, 否, zero_gradient_is_stationary_not_a_minimum_certificate)","explanation_zh":"Wave6第十三層核對=(zero_gradient_nonminimum, f(x)=-6*x^2, x=0, gradient=0, strict_local_maximum, 否, zero_gradient_is_stationary_not_a_minimum_certificate)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["梯度下降"]},"variant":{"family_id":"qf-mod50-gd-boundary","seed":202611193081,"parameters":{"mode":"zero_gradient_nonminimum","model":{"scale":6},"domain":"Exact_diagonal_quadratic_gradient_descent","operation":"gd-boundary","semantic_template":"gd-boundary|{\"mode\":\"zero_gradient_nonminimum\",\"model\":{\"scale\":[6,1]}}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r03-0082","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-descent"],"title_zh":"梯度下降","stem_zh":"standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-boundary；明示資料={\"mode\":\"finite_trace_only\",\"model\":{\"diagonal\":[1,2],\"linear\":[6,-6],\"point\":[7,1],\"step_size\":[1,4],\"steps\":1}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(finite_trace_only, 1, [[0, [7, 1], 123/2], [1, [15/4, 2], 689/32]], 否, finite_trace_does_not_prove_convergence_or_global_optimality)","explanation_zh":"Wave6第十三層核對=(finite_trace_only, 1, [[0, [7, 1], 123/2], [1, [15/4, 2], 689/32]], 否, finite_trace_does_not_prove_convergence_or_global_optimality)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["梯度下降"]},"variant":{"family_id":"qf-mod50-gd-boundary","seed":202611193082,"parameters":{"mode":"finite_trace_only","model":{"diagonal":[1,2],"linear":[6,-6],"point":[7,1],"step_size":[1,4],"steps":1},"domain":"Exact_diagonal_quadratic_gradient_descent","operation":"gd-boundary","semantic_template":"gd-boundary|{\"mode\":\"finite_trace_only\",\"model\":{\"diagonal\":[[1,1],[2,1]],\"linear\":[[6,1],[-6,1]],\"point\":[[7,1],[1,1]],\"step_size\":[1,4],\"steps\":1}}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r03-0083","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-descent"],"title_zh":"梯度下降","stem_zh":"standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-boundary；明示資料={\"mode\":\"constrained_confusion\",\"model\":{\"gradient\":8,\"lower\":-1,\"point\":0,\"step_size\":2,\"upper\":1}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(constrained_confusion, -1, 1, 0, -16, 否, unconstrained_gradient_step_is_not_projected_or_constrained_descent)","explanation_zh":"Wave6第十三層核對=(constrained_confusion, -1, 1, 0, -16, 否, unconstrained_gradient_step_is_not_projected_or_constrained_descent)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["梯度下降"]},"variant":{"family_id":"qf-mod50-gd-boundary","seed":202611193083,"parameters":{"mode":"constrained_confusion","model":{"point":0,"gradient":8,"step_size":2,"lower":-1,"upper":1},"domain":"Exact_diagonal_quadratic_gradient_descent","operation":"gd-boundary","semantic_template":"gd-boundary|{\"mode\":\"constrained_confusion\",\"model\":{\"gradient\":[8,1],\"lower\":[-1,1],\"point\":[0,1],\"step_size\":[2,1],\"upper\":[1,1]}}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r03-0084","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-descent"],"title_zh":"梯度下降","stem_zh":"standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-boundary；明示資料={\"mode\":\"oversized_step\",\"model\":{\"curvature\":4,\"point\":9,\"step_size\":3}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(oversized_step, 4, 9, 3, -99, 162, 19602, 否, positive_but_oversized_step_can_increase_the_objective)","explanation_zh":"Wave6第十三層核對=(oversized_step, 4, 9, 3, -99, 162, 19602, 否, positive_but_oversized_step_can_increase_the_objective)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度下降"]},"variant":{"family_id":"qf-mod50-gd-boundary","seed":202611193084,"parameters":{"mode":"oversized_step","model":{"curvature":4,"point":9,"step_size":3},"domain":"Exact_diagonal_quadratic_gradient_descent","operation":"gd-boundary","semantic_template":"gd-boundary|{\"mode\":\"oversized_step\",\"model\":{\"curvature\":[4,1],\"point\":[9,1],\"step_size\":[3,1]}}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r03-0085","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-descent"],"title_zh":"梯度下降","stem_zh":"standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-boundary；明示資料={\"mode\":\"zero_gradient_nonminimum\",\"model\":{\"scale\":10}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(zero_gradient_nonminimum, f(x)=-10*x^2, x=0, gradient=0, strict_local_maximum, 否, zero_gradient_is_stationary_not_a_minimum_certificate)","explanation_zh":"Wave6第十三層核對=(zero_gradient_nonminimum, f(x)=-10*x^2, x=0, gradient=0, strict_local_maximum, 否, zero_gradient_is_stationary_not_a_minimum_certificate)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度下降"]},"variant":{"family_id":"qf-mod50-gd-boundary","seed":202611193085,"parameters":{"mode":"zero_gradient_nonminimum","model":{"scale":10},"domain":"Exact_diagonal_quadratic_gradient_descent","operation":"gd-boundary","semantic_template":"gd-boundary|{\"mode\":\"zero_gradient_nonminimum\",\"model\":{\"scale\":[10,1]}}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r03-0086","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-descent"],"title_zh":"梯度下降","stem_zh":"standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-boundary；明示資料={\"mode\":\"finite_trace_only\",\"model\":{\"diagonal\":[1,2],\"linear\":[10,-10],\"point\":[11,1],\"step_size\":[1,4],\"steps\":1}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(finite_trace_only, 1, [[0, [11, 1], 323/2], [1, [23/4, 3], 1697/32]], 否, finite_trace_does_not_prove_convergence_or_global_optimality)","explanation_zh":"Wave6第十三層核對=(finite_trace_only, 1, [[0, [11, 1], 323/2], [1, [23/4, 3], 1697/32]], 否, finite_trace_does_not_prove_convergence_or_global_optimality)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度下降"]},"variant":{"family_id":"qf-mod50-gd-boundary","seed":202611193086,"parameters":{"mode":"finite_trace_only","model":{"diagonal":[1,2],"linear":[10,-10],"point":[11,1],"step_size":[1,4],"steps":1},"domain":"Exact_diagonal_quadratic_gradient_descent","operation":"gd-boundary","semantic_template":"gd-boundary|{\"mode\":\"finite_trace_only\",\"model\":{\"diagonal\":[[1,1],[2,1]],\"linear\":[[10,1],[-10,1]],\"point\":[[11,1],[1,1]],\"step_size\":[1,4],\"steps\":1}}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r03-0087","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-descent"],"title_zh":"梯度下降","stem_zh":"standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-boundary；明示資料={\"mode\":\"constrained_confusion\",\"model\":{\"gradient\":12,\"lower\":-1,\"point\":0,\"step_size\":2,\"upper\":1}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(constrained_confusion, -1, 1, 0, -24, 否, unconstrained_gradient_step_is_not_projected_or_constrained_descent)","explanation_zh":"Wave6第十三層核對=(constrained_confusion, -1, 1, 0, -24, 否, unconstrained_gradient_step_is_not_projected_or_constrained_descent)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度下降"]},"variant":{"family_id":"qf-mod50-gd-boundary","seed":202611193087,"parameters":{"mode":"constrained_confusion","model":{"point":0,"gradient":12,"step_size":2,"lower":-1,"upper":1},"domain":"Exact_diagonal_quadratic_gradient_descent","operation":"gd-boundary","semantic_template":"gd-boundary|{\"mode\":\"constrained_confusion\",\"model\":{\"gradient\":[12,1],\"lower\":[-1,1],\"point\":[0,1],\"step_size\":[2,1],\"upper\":[1,1]}}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r03-0088","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-descent"],"title_zh":"梯度下降","stem_zh":"standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-boundary；明示資料={\"mode\":\"oversized_step\",\"model\":{\"curvature\":3,\"point\":13,\"step_size\":3}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(oversized_step, 3, 13, 3, -104, 507/2, 16224, 否, positive_but_oversized_step_can_increase_the_objective)","explanation_zh":"Wave6第十三層核對=(oversized_step, 3, 13, 3, -104, 507/2, 16224, 否, positive_but_oversized_step_can_increase_the_objective)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度下降"]},"variant":{"family_id":"qf-mod50-gd-boundary","seed":202611193088,"parameters":{"mode":"oversized_step","model":{"curvature":3,"point":13,"step_size":3},"domain":"Exact_diagonal_quadratic_gradient_descent","operation":"gd-boundary","semantic_template":"gd-boundary|{\"mode\":\"oversized_step\",\"model\":{\"curvature\":[3,1],\"point\":[13,1],\"step_size\":[3,1]}}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r03-0089","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-descent"],"title_zh":"梯度下降","stem_zh":"standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-boundary；明示資料={\"mode\":\"zero_gradient_nonminimum\",\"model\":{\"scale\":14}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(zero_gradient_nonminimum, f(x)=-14*x^2, x=0, gradient=0, strict_local_maximum, 否, zero_gradient_is_stationary_not_a_minimum_certificate)","explanation_zh":"Wave6第十三層核對=(zero_gradient_nonminimum, f(x)=-14*x^2, x=0, gradient=0, strict_local_maximum, 否, zero_gradient_is_stationary_not_a_minimum_certificate)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度下降"]},"variant":{"family_id":"qf-mod50-gd-boundary","seed":202611193089,"parameters":{"mode":"zero_gradient_nonminimum","model":{"scale":14},"domain":"Exact_diagonal_quadratic_gradient_descent","operation":"gd-boundary","semantic_template":"gd-boundary|{\"mode\":\"zero_gradient_nonminimum\",\"model\":{\"scale\":[14,1]}}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r03-0090","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-descent"],"title_zh":"梯度下降","stem_zh":"standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-boundary；明示資料={\"mode\":\"finite_trace_only\",\"model\":{\"diagonal\":[1,2],\"linear\":[14,-14],\"point\":[15,1],\"step_size\":[1,4],\"steps\":1}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(finite_trace_only, 1, [[0, [15, 1], 619/2], [1, [31/4, 4], 3153/32]], 否, finite_trace_does_not_prove_convergence_or_global_optimality)","explanation_zh":"Wave6第十三層核對=(finite_trace_only, 1, [[0, [15, 1], 619/2], [1, [31/4, 4], 3153/32]], 否, finite_trace_does_not_prove_convergence_or_global_optimality)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度下降"]},"variant":{"family_id":"qf-mod50-gd-boundary","seed":202611193090,"parameters":{"mode":"finite_trace_only","model":{"diagonal":[1,2],"linear":[14,-14],"point":[15,1],"step_size":[1,4],"steps":1},"domain":"Exact_diagonal_quadratic_gradient_descent","operation":"gd-boundary","semantic_template":"gd-boundary|{\"mode\":\"finite_trace_only\",\"model\":{\"diagonal\":[[1,1],[2,1]],\"linear\":[[14,1],[-14,1]],\"point\":[[15,1],[1,1]],\"step_size\":[1,4],\"steps\":1}}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r03-0091","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-descent"],"title_zh":"梯度下降","stem_zh":"standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-boundary；明示資料={\"mode\":\"constrained_confusion\",\"model\":{\"gradient\":16,\"lower\":-1,\"point\":0,\"step_size\":2,\"upper\":1}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(constrained_confusion, -1, 1, 0, -32, 否, unconstrained_gradient_step_is_not_projected_or_constrained_descent)","explanation_zh":"Wave6第十三層核對=(constrained_confusion, -1, 1, 0, -32, 否, unconstrained_gradient_step_is_not_projected_or_constrained_descent)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度下降"]},"variant":{"family_id":"qf-mod50-gd-boundary","seed":202611193091,"parameters":{"mode":"constrained_confusion","model":{"point":0,"gradient":16,"step_size":2,"lower":-1,"upper":1},"domain":"Exact_diagonal_quadratic_gradient_descent","operation":"gd-boundary","semantic_template":"gd-boundary|{\"mode\":\"constrained_confusion\",\"model\":{\"gradient\":[16,1],\"lower\":[-1,1],\"point\":[0,1],\"step_size\":[2,1],\"upper\":[1,1]}}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r03-0092","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-descent"],"title_zh":"梯度下降","stem_zh":"standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-boundary；明示資料={\"mode\":\"oversized_step\",\"model\":{\"curvature\":2,\"point\":17,\"step_size\":3}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(oversized_step, 2, 17, 3, -85, 289, 7225, 否, positive_but_oversized_step_can_increase_the_objective)","explanation_zh":"Wave6第十三層核對=(oversized_step, 2, 17, 3, -85, 289, 7225, 否, positive_but_oversized_step_can_increase_the_objective)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度下降"]},"variant":{"family_id":"qf-mod50-gd-boundary","seed":202611193092,"parameters":{"mode":"oversized_step","model":{"curvature":2,"point":17,"step_size":3},"domain":"Exact_diagonal_quadratic_gradient_descent","operation":"gd-boundary","semantic_template":"gd-boundary|{\"mode\":\"oversized_step\",\"model\":{\"curvature\":[2,1],\"point\":[17,1],\"step_size\":[3,1]}}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r03-0093","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-descent"],"title_zh":"梯度下降","stem_zh":"standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-boundary；明示資料={\"mode\":\"zero_gradient_nonminimum\",\"model\":{\"scale\":18}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(zero_gradient_nonminimum, f(x)=-18*x^2, x=0, gradient=0, strict_local_maximum, 否, zero_gradient_is_stationary_not_a_minimum_certificate)","explanation_zh":"Wave6第十三層核對=(zero_gradient_nonminimum, f(x)=-18*x^2, x=0, gradient=0, strict_local_maximum, 否, zero_gradient_is_stationary_not_a_minimum_certificate)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["梯度下降"]},"variant":{"family_id":"qf-mod50-gd-boundary","seed":202611193093,"parameters":{"mode":"zero_gradient_nonminimum","model":{"scale":18},"domain":"Exact_diagonal_quadratic_gradient_descent","operation":"gd-boundary","semantic_template":"gd-boundary|{\"mode\":\"zero_gradient_nonminimum\",\"model\":{\"scale\":[18,1]}}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r03-0094","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-descent"],"title_zh":"梯度下降","stem_zh":"standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-boundary；明示資料={\"mode\":\"finite_trace_only\",\"model\":{\"diagonal\":[1,2],\"linear\":[18,-18],\"point\":[19,1],\"step_size\":[1,4],\"steps\":1}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(finite_trace_only, 1, [[0, [19, 1], 1011/2], [1, [39/4, 5], 5057/32]], 否, finite_trace_does_not_prove_convergence_or_global_optimality)","explanation_zh":"Wave6第十三層核對=(finite_trace_only, 1, [[0, [19, 1], 1011/2], [1, [39/4, 5], 5057/32]], 否, finite_trace_does_not_prove_convergence_or_global_optimality)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["梯度下降"]},"variant":{"family_id":"qf-mod50-gd-boundary","seed":202611193094,"parameters":{"mode":"finite_trace_only","model":{"diagonal":[1,2],"linear":[18,-18],"point":[19,1],"step_size":[1,4],"steps":1},"domain":"Exact_diagonal_quadratic_gradient_descent","operation":"gd-boundary","semantic_template":"gd-boundary|{\"mode\":\"finite_trace_only\",\"model\":{\"diagonal\":[[1,1],[2,1]],\"linear\":[[18,1],[-18,1]],\"point\":[[19,1],[1,1]],\"step_size\":[1,4],\"steps\":1}}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r03-0095","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-descent"],"title_zh":"梯度下降","stem_zh":"standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-boundary；明示資料={\"mode\":\"constrained_confusion\",\"model\":{\"gradient\":20,\"lower\":-1,\"point\":0,\"step_size\":2,\"upper\":1}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(constrained_confusion, -1, 1, 0, -40, 否, unconstrained_gradient_step_is_not_projected_or_constrained_descent)","explanation_zh":"Wave6第十三層核對=(constrained_confusion, -1, 1, 0, -40, 否, unconstrained_gradient_step_is_not_projected_or_constrained_descent)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["梯度下降"]},"variant":{"family_id":"qf-mod50-gd-boundary","seed":202611193095,"parameters":{"mode":"constrained_confusion","model":{"point":0,"gradient":20,"step_size":2,"lower":-1,"upper":1},"domain":"Exact_diagonal_quadratic_gradient_descent","operation":"gd-boundary","semantic_template":"gd-boundary|{\"mode\":\"constrained_confusion\",\"model\":{\"gradient\":[20,1],\"lower\":[-1,1],\"point\":[0,1],\"step_size\":[2,1],\"upper\":[1,1]}}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r03-0096","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-descent"],"title_zh":"梯度下降","stem_zh":"standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-boundary；明示資料={\"mode\":\"oversized_step\",\"model\":{\"curvature\":1,\"point\":21,\"step_size\":3}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(oversized_step, 1, 21, 3, -42, 441/2, 882, 否, positive_but_oversized_step_can_increase_the_objective)","explanation_zh":"Wave6第十三層核對=(oversized_step, 1, 21, 3, -42, 441/2, 882, 否, positive_but_oversized_step_can_increase_the_objective)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["梯度下降"]},"variant":{"family_id":"qf-mod50-gd-boundary","seed":202611193096,"parameters":{"mode":"oversized_step","model":{"curvature":1,"point":21,"step_size":3},"domain":"Exact_diagonal_quadratic_gradient_descent","operation":"gd-boundary","semantic_template":"gd-boundary|{\"mode\":\"oversized_step\",\"model\":{\"curvature\":[1,1],\"point\":[21,1],\"step_size\":[3,1]}}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r03-0097","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-descent"],"title_zh":"梯度下降","stem_zh":"standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-boundary；明示資料={\"mode\":\"zero_gradient_nonminimum\",\"model\":{\"scale\":22}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(zero_gradient_nonminimum, f(x)=-22*x^2, x=0, gradient=0, strict_local_maximum, 否, zero_gradient_is_stationary_not_a_minimum_certificate)","explanation_zh":"Wave6第十三層核對=(zero_gradient_nonminimum, f(x)=-22*x^2, x=0, gradient=0, strict_local_maximum, 否, zero_gradient_is_stationary_not_a_minimum_certificate)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["梯度下降"]},"variant":{"family_id":"qf-mod50-gd-boundary","seed":202611193097,"parameters":{"mode":"zero_gradient_nonminimum","model":{"scale":22},"domain":"Exact_diagonal_quadratic_gradient_descent","operation":"gd-boundary","semantic_template":"gd-boundary|{\"mode\":\"zero_gradient_nonminimum\",\"model\":{\"scale\":[22,1]}}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r03-0098","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-descent"],"title_zh":"梯度下降","stem_zh":"standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-boundary；明示資料={\"mode\":\"finite_trace_only\",\"model\":{\"diagonal\":[1,2],\"linear\":[22,-22],\"point\":[23,1],\"step_size\":[1,4],\"steps\":1}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(finite_trace_only, 1, [[0, [23, 1], 1499/2], [1, [47/4, 6], 7409/32]], 否, finite_trace_does_not_prove_convergence_or_global_optimality)","explanation_zh":"Wave6第十三層核對=(finite_trace_only, 1, [[0, [23, 1], 1499/2], [1, [47/4, 6], 7409/32]], 否, finite_trace_does_not_prove_convergence_or_global_optimality)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["梯度下降"]},"variant":{"family_id":"qf-mod50-gd-boundary","seed":202611193098,"parameters":{"mode":"finite_trace_only","model":{"diagonal":[1,2],"linear":[22,-22],"point":[23,1],"step_size":[1,4],"steps":1},"domain":"Exact_diagonal_quadratic_gradient_descent","operation":"gd-boundary","semantic_template":"gd-boundary|{\"mode\":\"finite_trace_only\",\"model\":{\"diagonal\":[[1,1],[2,1]],\"linear\":[[22,1],[-22,1]],\"point\":[[23,1],[1,1]],\"step_size\":[1,4],\"steps\":1}}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r03-0099","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-descent"],"title_zh":"梯度下降","stem_zh":"standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-boundary；明示資料={\"mode\":\"constrained_confusion\",\"model\":{\"gradient\":24,\"lower\":-1,\"point\":0,\"step_size\":2,\"upper\":1}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(constrained_confusion, -1, 1, 0, -48, 否, unconstrained_gradient_step_is_not_projected_or_constrained_descent)","explanation_zh":"Wave6第十三層核對=(constrained_confusion, -1, 1, 0, -48, 否, unconstrained_gradient_step_is_not_projected_or_constrained_descent)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["梯度下降"]},"variant":{"family_id":"qf-mod50-gd-boundary","seed":202611193099,"parameters":{"mode":"constrained_confusion","model":{"point":0,"gradient":24,"step_size":2,"lower":-1,"upper":1},"domain":"Exact_diagonal_quadratic_gradient_descent","operation":"gd-boundary","semantic_template":"gd-boundary|{\"mode\":\"constrained_confusion\",\"model\":{\"gradient\":[24,1],\"lower\":[-1,1],\"point\":[0,1],\"step_size\":[2,1],\"upper\":[1,1]}}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r03-0100","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-descent"],"title_zh":"梯度下降","stem_zh":"standard Euclidean coordinate order、正步長與完整gradient update。操作=gd-boundary；明示資料={\"mode\":\"oversized_step\",\"model\":{\"curvature\":5,\"point\":25,\"step_size\":3}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(oversized_step, 5, 25, 3, -350, 3125/2, 306250, 否, positive_but_oversized_step_can_increase_the_objective)","explanation_zh":"Wave6第十三層核對=(oversized_step, 5, 25, 3, -350, 3125/2, 306250, 否, positive_but_oversized_step_can_increase_the_objective)；依完整明示資料精確重算；x_next=x-alpha*gradient逐座標精確重算；safe trace另核0<alpha<2/L；positive alpha、zero gradient或finite trace不自動給descent、minimum、global optimum或convergence；不冒充projected descent。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["梯度下降"]},"variant":{"family_id":"qf-mod50-gd-boundary","seed":202611193100,"parameters":{"mode":"oversized_step","model":{"curvature":5,"point":25,"step_size":3},"domain":"Exact_diagonal_quadratic_gradient_descent","operation":"gd-boundary","semantic_template":"gd-boundary|{\"mode\":\"oversized_step\",\"model\":{\"curvature\":[5,1],\"point\":[25,1],\"step_size\":[3,1]}}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}