{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r06-0001","batch_id":"batch-20261015-advance-20261119-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-linearity；明示資料={\"alpha\":[1,2],\"beta\":[1,3],\"gradient_f\":[1,0],\"gradient_g\":[0,-1]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([1, 0], [0, -1], 1/2, 1/3, [1/2, -1/3], pointwise_gradient_linearity_in_fixed_coordinates)","explanation_zh":"Wave6第十三層核對=([1, 0], [0, -1], 1/2, 1/3, [1/2, -1/3], pointwise_gradient_linearity_in_fixed_coordinates)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod50-graddeep-linearity","seed":202611196001,"parameters":{"gradient_f":[1,0],"gradient_g":[0,-1],"alpha":[1,2],"beta":[1,3],"domain":"Exact_gradient_vector_density","operation":"graddeep-linearity","semantic_template":"graddeep-linearity|{\"alpha\":[1,2],\"beta\":[1,3],\"gradient_f\":[[1,1],[0,1]],\"gradient_g\":[[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-r06-0002","batch_id":"batch-20261015-advance-20261119-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-linearity；明示資料={\"alpha\":[1,2],\"beta\":[2,3],\"gradient_f\":[2,1],\"gradient_g\":[1,-2]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([2, 1], [1, -2], 1/2, 2/3, [5/3, -5/6], pointwise_gradient_linearity_in_fixed_coordinates)","explanation_zh":"Wave6第十三層核對=([2, 1], [1, -2], 1/2, 2/3, [5/3, -5/6], pointwise_gradient_linearity_in_fixed_coordinates)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod50-graddeep-linearity","seed":202611196002,"parameters":{"gradient_f":[2,1],"gradient_g":[1,-2],"alpha":[1,2],"beta":[2,3],"domain":"Exact_gradient_vector_density","operation":"graddeep-linearity","semantic_template":"graddeep-linearity|{\"alpha\":[1,2],\"beta\":[2,3],\"gradient_f\":[[2,1],[1,1]],\"gradient_g\":[[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-r06-0003","batch_id":"batch-20261015-advance-20261119-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-linearity；明示資料={\"alpha\":[1,2],\"beta\":[3,3],\"gradient_f\":[3,2],\"gradient_g\":[2,-3]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([3, 2], [2, -3], 1/2, 1, [7/2, -2], pointwise_gradient_linearity_in_fixed_coordinates)","explanation_zh":"Wave6第十三層核對=([3, 2], [2, -3], 1/2, 1, [7/2, -2], pointwise_gradient_linearity_in_fixed_coordinates)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod50-graddeep-linearity","seed":202611196003,"parameters":{"gradient_f":[3,2],"gradient_g":[2,-3],"alpha":[1,2],"beta":[3,3],"domain":"Exact_gradient_vector_density","operation":"graddeep-linearity","semantic_template":"graddeep-linearity|{\"alpha\":[1,2],\"beta\":[1,1],\"gradient_f\":[[3,1],[2,1]],\"gradient_g\":[[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-r06-0004","batch_id":"batch-20261015-advance-20261119-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-linearity；明示資料={\"alpha\":[1,2],\"beta\":[4,3],\"gradient_f\":[4,3],\"gradient_g\":[3,-4]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([4, 3], [3, -4], 1/2, 4/3, [6, -23/6], pointwise_gradient_linearity_in_fixed_coordinates)","explanation_zh":"Wave6第十三層核對=([4, 3], [3, -4], 1/2, 4/3, [6, -23/6], pointwise_gradient_linearity_in_fixed_coordinates)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod50-graddeep-linearity","seed":202611196004,"parameters":{"gradient_f":[4,3],"gradient_g":[3,-4],"alpha":[1,2],"beta":[4,3],"domain":"Exact_gradient_vector_density","operation":"graddeep-linearity","semantic_template":"graddeep-linearity|{\"alpha\":[1,2],\"beta\":[4,3],\"gradient_f\":[[4,1],[3,1]],\"gradient_g\":[[3,1],[-4,1]]}"}},"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-r06-0005","batch_id":"batch-20261015-advance-20261119-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-linearity；明示資料={\"alpha\":[1,2],\"beta\":[1,3],\"gradient_f\":[5,4],\"gradient_g\":[4,-5]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([5, 4], [4, -5], 1/2, 1/3, [23/6, 1/3], pointwise_gradient_linearity_in_fixed_coordinates)","explanation_zh":"Wave6第十三層核對=([5, 4], [4, -5], 1/2, 1/3, [23/6, 1/3], pointwise_gradient_linearity_in_fixed_coordinates)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod50-graddeep-linearity","seed":202611196005,"parameters":{"gradient_f":[5,4],"gradient_g":[4,-5],"alpha":[1,2],"beta":[1,3],"domain":"Exact_gradient_vector_density","operation":"graddeep-linearity","semantic_template":"graddeep-linearity|{\"alpha\":[1,2],\"beta\":[1,3],\"gradient_f\":[[5,1],[4,1]],\"gradient_g\":[[4,1],[-5,1]]}"}},"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-r06-0006","batch_id":"batch-20261015-advance-20261119-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-linearity；明示資料={\"alpha\":[1,2],\"beta\":[2,3],\"gradient_f\":[6,0],\"gradient_g\":[5,-6]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([6, 0], [5, -6], 1/2, 2/3, [19/3, -4], pointwise_gradient_linearity_in_fixed_coordinates)","explanation_zh":"Wave6第十三層核對=([6, 0], [5, -6], 1/2, 2/3, [19/3, -4], pointwise_gradient_linearity_in_fixed_coordinates)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod50-graddeep-linearity","seed":202611196006,"parameters":{"gradient_f":[6,0],"gradient_g":[5,-6],"alpha":[1,2],"beta":[2,3],"domain":"Exact_gradient_vector_density","operation":"graddeep-linearity","semantic_template":"graddeep-linearity|{\"alpha\":[1,2],\"beta\":[2,3],\"gradient_f\":[[6,1],[0,1]],\"gradient_g\":[[5,1],[-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-r06-0007","batch_id":"batch-20261015-advance-20261119-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-linearity；明示資料={\"alpha\":[1,2],\"beta\":[3,3],\"gradient_f\":[7,1],\"gradient_g\":[6,-7]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([7, 1], [6, -7], 1/2, 1, [19/2, -13/2], pointwise_gradient_linearity_in_fixed_coordinates)","explanation_zh":"Wave6第十三層核對=([7, 1], [6, -7], 1/2, 1, [19/2, -13/2], pointwise_gradient_linearity_in_fixed_coordinates)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod50-graddeep-linearity","seed":202611196007,"parameters":{"gradient_f":[7,1],"gradient_g":[6,-7],"alpha":[1,2],"beta":[3,3],"domain":"Exact_gradient_vector_density","operation":"graddeep-linearity","semantic_template":"graddeep-linearity|{\"alpha\":[1,2],\"beta\":[1,1],\"gradient_f\":[[7,1],[1,1]],\"gradient_g\":[[6,1],[-7,1]]}"}},"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-r06-0008","batch_id":"batch-20261015-advance-20261119-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-linearity；明示資料={\"alpha\":[1,2],\"beta\":[4,3],\"gradient_f\":[8,2],\"gradient_g\":[0,-8]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([8, 2], [0, -8], 1/2, 4/3, [4, -29/3], pointwise_gradient_linearity_in_fixed_coordinates)","explanation_zh":"Wave6第十三層核對=([8, 2], [0, -8], 1/2, 4/3, [4, -29/3], pointwise_gradient_linearity_in_fixed_coordinates)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod50-graddeep-linearity","seed":202611196008,"parameters":{"gradient_f":[8,2],"gradient_g":[0,-8],"alpha":[1,2],"beta":[4,3],"domain":"Exact_gradient_vector_density","operation":"graddeep-linearity","semantic_template":"graddeep-linearity|{\"alpha\":[1,2],\"beta\":[4,3],\"gradient_f\":[[8,1],[2,1]],\"gradient_g\":[[0,1],[-8,1]]}"}},"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-r06-0009","batch_id":"batch-20261015-advance-20261119-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-linearity；明示資料={\"alpha\":[1,2],\"beta\":[1,3],\"gradient_f\":[9,3],\"gradient_g\":[1,-9]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([9, 3], [1, -9], 1/2, 1/3, [29/6, -3/2], pointwise_gradient_linearity_in_fixed_coordinates)","explanation_zh":"Wave6第十三層核對=([9, 3], [1, -9], 1/2, 1/3, [29/6, -3/2], pointwise_gradient_linearity_in_fixed_coordinates)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod50-graddeep-linearity","seed":202611196009,"parameters":{"gradient_f":[9,3],"gradient_g":[1,-9],"alpha":[1,2],"beta":[1,3],"domain":"Exact_gradient_vector_density","operation":"graddeep-linearity","semantic_template":"graddeep-linearity|{\"alpha\":[1,2],\"beta\":[1,3],\"gradient_f\":[[9,1],[3,1]],\"gradient_g\":[[1,1],[-9,1]]}"}},"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-r06-0010","batch_id":"batch-20261015-advance-20261119-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-linearity；明示資料={\"alpha\":[1,2],\"beta\":[2,3],\"gradient_f\":[10,4],\"gradient_g\":[2,-10]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([10, 4], [2, -10], 1/2, 2/3, [19/3, -14/3], pointwise_gradient_linearity_in_fixed_coordinates)","explanation_zh":"Wave6第十三層核對=([10, 4], [2, -10], 1/2, 2/3, [19/3, -14/3], pointwise_gradient_linearity_in_fixed_coordinates)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod50-graddeep-linearity","seed":202611196010,"parameters":{"gradient_f":[10,4],"gradient_g":[2,-10],"alpha":[1,2],"beta":[2,3],"domain":"Exact_gradient_vector_density","operation":"graddeep-linearity","semantic_template":"graddeep-linearity|{\"alpha\":[1,2],\"beta\":[2,3],\"gradient_f\":[[10,1],[4,1]],\"gradient_g\":[[2,1],[-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-r06-0011","batch_id":"batch-20261015-advance-20261119-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-linearity；明示資料={\"alpha\":[1,2],\"beta\":[3,3],\"gradient_f\":[11,0],\"gradient_g\":[3,-11]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([11, 0], [3, -11], 1/2, 1, [17/2, -11], pointwise_gradient_linearity_in_fixed_coordinates)","explanation_zh":"Wave6第十三層核對=([11, 0], [3, -11], 1/2, 1, [17/2, -11], pointwise_gradient_linearity_in_fixed_coordinates)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod50-graddeep-linearity","seed":202611196011,"parameters":{"gradient_f":[11,0],"gradient_g":[3,-11],"alpha":[1,2],"beta":[3,3],"domain":"Exact_gradient_vector_density","operation":"graddeep-linearity","semantic_template":"graddeep-linearity|{\"alpha\":[1,2],\"beta\":[1,1],\"gradient_f\":[[11,1],[0,1]],\"gradient_g\":[[3,1],[-11,1]]}"}},"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-r06-0012","batch_id":"batch-20261015-advance-20261119-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-linearity；明示資料={\"alpha\":[1,2],\"beta\":[4,3],\"gradient_f\":[12,1],\"gradient_g\":[4,-12]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([12, 1], [4, -12], 1/2, 4/3, [34/3, -31/2], pointwise_gradient_linearity_in_fixed_coordinates)","explanation_zh":"Wave6第十三層核對=([12, 1], [4, -12], 1/2, 4/3, [34/3, -31/2], pointwise_gradient_linearity_in_fixed_coordinates)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod50-graddeep-linearity","seed":202611196012,"parameters":{"gradient_f":[12,1],"gradient_g":[4,-12],"alpha":[1,2],"beta":[4,3],"domain":"Exact_gradient_vector_density","operation":"graddeep-linearity","semantic_template":"graddeep-linearity|{\"alpha\":[1,2],\"beta\":[4,3],\"gradient_f\":[[12,1],[1,1]],\"gradient_g\":[[4,1],[-12,1]]}"}},"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-r06-0013","batch_id":"batch-20261015-advance-20261119-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-linearity；明示資料={\"alpha\":[1,2],\"beta\":[1,3],\"gradient_f\":[13,2],\"gradient_g\":[5,-13]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([13, 2], [5, -13], 1/2, 1/3, [49/6, -10/3], pointwise_gradient_linearity_in_fixed_coordinates)","explanation_zh":"Wave6第十三層核對=([13, 2], [5, -13], 1/2, 1/3, [49/6, -10/3], pointwise_gradient_linearity_in_fixed_coordinates)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod50-graddeep-linearity","seed":202611196013,"parameters":{"gradient_f":[13,2],"gradient_g":[5,-13],"alpha":[1,2],"beta":[1,3],"domain":"Exact_gradient_vector_density","operation":"graddeep-linearity","semantic_template":"graddeep-linearity|{\"alpha\":[1,2],\"beta\":[1,3],\"gradient_f\":[[13,1],[2,1]],\"gradient_g\":[[5,1],[-13,1]]}"}},"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-r06-0014","batch_id":"batch-20261015-advance-20261119-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-linearity；明示資料={\"alpha\":[1,2],\"beta\":[2,3],\"gradient_f\":[14,3],\"gradient_g\":[6,-14]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([14, 3], [6, -14], 1/2, 2/3, [11, -47/6], pointwise_gradient_linearity_in_fixed_coordinates)","explanation_zh":"Wave6第十三層核對=([14, 3], [6, -14], 1/2, 2/3, [11, -47/6], pointwise_gradient_linearity_in_fixed_coordinates)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod50-graddeep-linearity","seed":202611196014,"parameters":{"gradient_f":[14,3],"gradient_g":[6,-14],"alpha":[1,2],"beta":[2,3],"domain":"Exact_gradient_vector_density","operation":"graddeep-linearity","semantic_template":"graddeep-linearity|{\"alpha\":[1,2],\"beta\":[2,3],\"gradient_f\":[[14,1],[3,1]],\"gradient_g\":[[6,1],[-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-r06-0015","batch_id":"batch-20261015-advance-20261119-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-linearity；明示資料={\"alpha\":[1,2],\"beta\":[3,3],\"gradient_f\":[15,4],\"gradient_g\":[0,-15]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([15, 4], [0, -15], 1/2, 1, [15/2, -13], pointwise_gradient_linearity_in_fixed_coordinates)","explanation_zh":"Wave6第十三層核對=([15, 4], [0, -15], 1/2, 1, [15/2, -13], pointwise_gradient_linearity_in_fixed_coordinates)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod50-graddeep-linearity","seed":202611196015,"parameters":{"gradient_f":[15,4],"gradient_g":[0,-15],"alpha":[1,2],"beta":[3,3],"domain":"Exact_gradient_vector_density","operation":"graddeep-linearity","semantic_template":"graddeep-linearity|{\"alpha\":[1,2],\"beta\":[1,1],\"gradient_f\":[[15,1],[4,1]],\"gradient_g\":[[0,1],[-15,1]]}"}},"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-r06-0016","batch_id":"batch-20261015-advance-20261119-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-linearity；明示資料={\"alpha\":[1,2],\"beta\":[4,3],\"gradient_f\":[16,0],\"gradient_g\":[1,-16]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([16, 0], [1, -16], 1/2, 4/3, [28/3, -64/3], pointwise_gradient_linearity_in_fixed_coordinates)","explanation_zh":"Wave6第十三層核對=([16, 0], [1, -16], 1/2, 4/3, [28/3, -64/3], pointwise_gradient_linearity_in_fixed_coordinates)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod50-graddeep-linearity","seed":202611196016,"parameters":{"gradient_f":[16,0],"gradient_g":[1,-16],"alpha":[1,2],"beta":[4,3],"domain":"Exact_gradient_vector_density","operation":"graddeep-linearity","semantic_template":"graddeep-linearity|{\"alpha\":[1,2],\"beta\":[4,3],\"gradient_f\":[[16,1],[0,1]],\"gradient_g\":[[1,1],[-16,1]]}"}},"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-r06-0017","batch_id":"batch-20261015-advance-20261119-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-linearity；明示資料={\"alpha\":[1,2],\"beta\":[1,3],\"gradient_f\":[17,1],\"gradient_g\":[2,-17]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([17, 1], [2, -17], 1/2, 1/3, [55/6, -31/6], pointwise_gradient_linearity_in_fixed_coordinates)","explanation_zh":"Wave6第十三層核對=([17, 1], [2, -17], 1/2, 1/3, [55/6, -31/6], pointwise_gradient_linearity_in_fixed_coordinates)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod50-graddeep-linearity","seed":202611196017,"parameters":{"gradient_f":[17,1],"gradient_g":[2,-17],"alpha":[1,2],"beta":[1,3],"domain":"Exact_gradient_vector_density","operation":"graddeep-linearity","semantic_template":"graddeep-linearity|{\"alpha\":[1,2],\"beta\":[1,3],\"gradient_f\":[[17,1],[1,1]],\"gradient_g\":[[2,1],[-17,1]]}"}},"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-r06-0018","batch_id":"batch-20261015-advance-20261119-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-linearity；明示資料={\"alpha\":[1,2],\"beta\":[2,3],\"gradient_f\":[18,2],\"gradient_g\":[3,-18]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([18, 2], [3, -18], 1/2, 2/3, [11, -11], pointwise_gradient_linearity_in_fixed_coordinates)","explanation_zh":"Wave6第十三層核對=([18, 2], [3, -18], 1/2, 2/3, [11, -11], pointwise_gradient_linearity_in_fixed_coordinates)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod50-graddeep-linearity","seed":202611196018,"parameters":{"gradient_f":[18,2],"gradient_g":[3,-18],"alpha":[1,2],"beta":[2,3],"domain":"Exact_gradient_vector_density","operation":"graddeep-linearity","semantic_template":"graddeep-linearity|{\"alpha\":[1,2],\"beta\":[2,3],\"gradient_f\":[[18,1],[2,1]],\"gradient_g\":[[3,1],[-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-r06-0019","batch_id":"batch-20261015-advance-20261119-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-linearity；明示資料={\"alpha\":[1,2],\"beta\":[3,3],\"gradient_f\":[19,3],\"gradient_g\":[4,-19]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([19, 3], [4, -19], 1/2, 1, [27/2, -35/2], pointwise_gradient_linearity_in_fixed_coordinates)","explanation_zh":"Wave6第十三層核對=([19, 3], [4, -19], 1/2, 1, [27/2, -35/2], pointwise_gradient_linearity_in_fixed_coordinates)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod50-graddeep-linearity","seed":202611196019,"parameters":{"gradient_f":[19,3],"gradient_g":[4,-19],"alpha":[1,2],"beta":[3,3],"domain":"Exact_gradient_vector_density","operation":"graddeep-linearity","semantic_template":"graddeep-linearity|{\"alpha\":[1,2],\"beta\":[1,1],\"gradient_f\":[[19,1],[3,1]],\"gradient_g\":[[4,1],[-19,1]]}"}},"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-r06-0020","batch_id":"batch-20261015-advance-20261119-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-linearity；明示資料={\"alpha\":[1,2],\"beta\":[4,3],\"gradient_f\":[20,4],\"gradient_g\":[5,-20]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([20, 4], [5, -20], 1/2, 4/3, [50/3, -74/3], pointwise_gradient_linearity_in_fixed_coordinates)","explanation_zh":"Wave6第十三層核對=([20, 4], [5, -20], 1/2, 4/3, [50/3, -74/3], pointwise_gradient_linearity_in_fixed_coordinates)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod50-graddeep-linearity","seed":202611196020,"parameters":{"gradient_f":[20,4],"gradient_g":[5,-20],"alpha":[1,2],"beta":[4,3],"domain":"Exact_gradient_vector_density","operation":"graddeep-linearity","semantic_template":"graddeep-linearity|{\"alpha\":[1,2],\"beta\":[4,3],\"gradient_f\":[[20,1],[4,1]],\"gradient_g\":[[5,1],[-20,1]]}"}},"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-r06-0021","batch_id":"batch-20261015-advance-20261119-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-linearity；明示資料={\"alpha\":[1,2],\"beta\":[1,3],\"gradient_f\":[21,0],\"gradient_g\":[6,-21]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([21, 0], [6, -21], 1/2, 1/3, [25/2, -7], pointwise_gradient_linearity_in_fixed_coordinates)","explanation_zh":"Wave6第十三層核對=([21, 0], [6, -21], 1/2, 1/3, [25/2, -7], pointwise_gradient_linearity_in_fixed_coordinates)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod50-graddeep-linearity","seed":202611196021,"parameters":{"gradient_f":[21,0],"gradient_g":[6,-21],"alpha":[1,2],"beta":[1,3],"domain":"Exact_gradient_vector_density","operation":"graddeep-linearity","semantic_template":"graddeep-linearity|{\"alpha\":[1,2],\"beta\":[1,3],\"gradient_f\":[[21,1],[0,1]],\"gradient_g\":[[6,1],[-21,1]]}"}},"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-r06-0022","batch_id":"batch-20261015-advance-20261119-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-linearity；明示資料={\"alpha\":[1,2],\"beta\":[2,3],\"gradient_f\":[22,1],\"gradient_g\":[0,-22]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([22, 1], [0, -22], 1/2, 2/3, [11, -85/6], pointwise_gradient_linearity_in_fixed_coordinates)","explanation_zh":"Wave6第十三層核對=([22, 1], [0, -22], 1/2, 2/3, [11, -85/6], pointwise_gradient_linearity_in_fixed_coordinates)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod50-graddeep-linearity","seed":202611196022,"parameters":{"gradient_f":[22,1],"gradient_g":[0,-22],"alpha":[1,2],"beta":[2,3],"domain":"Exact_gradient_vector_density","operation":"graddeep-linearity","semantic_template":"graddeep-linearity|{\"alpha\":[1,2],\"beta\":[2,3],\"gradient_f\":[[22,1],[1,1]],\"gradient_g\":[[0,1],[-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-r06-0023","batch_id":"batch-20261015-advance-20261119-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-linearity；明示資料={\"alpha\":[1,2],\"beta\":[3,3],\"gradient_f\":[23,2],\"gradient_g\":[1,-23]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([23, 2], [1, -23], 1/2, 1, [25/2, -22], pointwise_gradient_linearity_in_fixed_coordinates)","explanation_zh":"Wave6第十三層核對=([23, 2], [1, -23], 1/2, 1, [25/2, -22], pointwise_gradient_linearity_in_fixed_coordinates)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod50-graddeep-linearity","seed":202611196023,"parameters":{"gradient_f":[23,2],"gradient_g":[1,-23],"alpha":[1,2],"beta":[3,3],"domain":"Exact_gradient_vector_density","operation":"graddeep-linearity","semantic_template":"graddeep-linearity|{\"alpha\":[1,2],\"beta\":[1,1],\"gradient_f\":[[23,1],[2,1]],\"gradient_g\":[[1,1],[-23,1]]}"}},"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-r06-0024","batch_id":"batch-20261015-advance-20261119-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-linearity；明示資料={\"alpha\":[1,2],\"beta\":[4,3],\"gradient_f\":[24,3],\"gradient_g\":[2,-24]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([24, 3], [2, -24], 1/2, 4/3, [44/3, -61/2], pointwise_gradient_linearity_in_fixed_coordinates)","explanation_zh":"Wave6第十三層核對=([24, 3], [2, -24], 1/2, 4/3, [44/3, -61/2], pointwise_gradient_linearity_in_fixed_coordinates)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod50-graddeep-linearity","seed":202611196024,"parameters":{"gradient_f":[24,3],"gradient_g":[2,-24],"alpha":[1,2],"beta":[4,3],"domain":"Exact_gradient_vector_density","operation":"graddeep-linearity","semantic_template":"graddeep-linearity|{\"alpha\":[1,2],\"beta\":[4,3],\"gradient_f\":[[24,1],[3,1]],\"gradient_g\":[[2,1],[-24,1]]}"}},"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-r06-0025","batch_id":"batch-20261015-advance-20261119-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-linearity；明示資料={\"alpha\":[1,2],\"beta\":[1,3],\"gradient_f\":[25,4],\"gradient_g\":[3,-25]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([25, 4], [3, -25], 1/2, 1/3, [27/2, -19/3], pointwise_gradient_linearity_in_fixed_coordinates)","explanation_zh":"Wave6第十三層核對=([25, 4], [3, -25], 1/2, 1/3, [27/2, -19/3], pointwise_gradient_linearity_in_fixed_coordinates)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod50-graddeep-linearity","seed":202611196025,"parameters":{"gradient_f":[25,4],"gradient_g":[3,-25],"alpha":[1,2],"beta":[1,3],"domain":"Exact_gradient_vector_density","operation":"graddeep-linearity","semantic_template":"graddeep-linearity|{\"alpha\":[1,2],\"beta\":[1,3],\"gradient_f\":[[25,1],[4,1]],\"gradient_g\":[[3,1],[-25,1]]}"}},"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-r06-0026","batch_id":"batch-20261015-advance-20261119-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-negative-direction；明示資料={\"gradient\":[1,-1]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([1, -1], [-1, 1], -2, 2, 是, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)","explanation_zh":"Wave6第十三層核對=([1, -1], [-1, 1], -2, 2, 是, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod50-graddeep-negative-direction","seed":202611196026,"parameters":{"gradient":[1,-1],"domain":"Exact_gradient_vector_density","operation":"graddeep-negative-direction","semantic_template":"graddeep-negative-direction|{\"gradient\":[[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-r06-0027","batch_id":"batch-20261015-advance-20261119-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-negative-direction；明示資料={\"gradient\":[2,-2]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([2, -2], [-2, 2], -8, 8, 是, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)","explanation_zh":"Wave6第十三層核對=([2, -2], [-2, 2], -8, 8, 是, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod50-graddeep-negative-direction","seed":202611196027,"parameters":{"gradient":[2,-2],"domain":"Exact_gradient_vector_density","operation":"graddeep-negative-direction","semantic_template":"graddeep-negative-direction|{\"gradient\":[[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-r06-0028","batch_id":"batch-20261015-advance-20261119-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-negative-direction；明示資料={\"gradient\":[3,-3]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([3, -3], [-3, 3], -18, 18, 是, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)","explanation_zh":"Wave6第十三層核對=([3, -3], [-3, 3], -18, 18, 是, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod50-graddeep-negative-direction","seed":202611196028,"parameters":{"gradient":[3,-3],"domain":"Exact_gradient_vector_density","operation":"graddeep-negative-direction","semantic_template":"graddeep-negative-direction|{\"gradient\":[[3,1],[-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-r06-0029","batch_id":"batch-20261015-advance-20261119-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-negative-direction；明示資料={\"gradient\":[4,-4]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([4, -4], [-4, 4], -32, 32, 是, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)","explanation_zh":"Wave6第十三層核對=([4, -4], [-4, 4], -32, 32, 是, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod50-graddeep-negative-direction","seed":202611196029,"parameters":{"gradient":[4,-4],"domain":"Exact_gradient_vector_density","operation":"graddeep-negative-direction","semantic_template":"graddeep-negative-direction|{\"gradient\":[[4,1],[-4,1]]}"}},"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-r06-0030","batch_id":"batch-20261015-advance-20261119-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-negative-direction；明示資料={\"gradient\":[5,-5]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([5, -5], [-5, 5], -50, 50, 是, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)","explanation_zh":"Wave6第十三層核對=([5, -5], [-5, 5], -50, 50, 是, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod50-graddeep-negative-direction","seed":202611196030,"parameters":{"gradient":[5,-5],"domain":"Exact_gradient_vector_density","operation":"graddeep-negative-direction","semantic_template":"graddeep-negative-direction|{\"gradient\":[[5,1],[-5,1]]}"}},"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-r06-0031","batch_id":"batch-20261015-advance-20261119-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-negative-direction；明示資料={\"gradient\":[6,-6]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([6, -6], [-6, 6], -72, 72, 是, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)","explanation_zh":"Wave6第十三層核對=([6, -6], [-6, 6], -72, 72, 是, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod50-graddeep-negative-direction","seed":202611196031,"parameters":{"gradient":[6,-6],"domain":"Exact_gradient_vector_density","operation":"graddeep-negative-direction","semantic_template":"graddeep-negative-direction|{\"gradient\":[[6,1],[-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-r06-0032","batch_id":"batch-20261015-advance-20261119-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-negative-direction；明示資料={\"gradient\":[7,-7]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([7, -7], [-7, 7], -98, 98, 是, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)","explanation_zh":"Wave6第十三層核對=([7, -7], [-7, 7], -98, 98, 是, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod50-graddeep-negative-direction","seed":202611196032,"parameters":{"gradient":[7,-7],"domain":"Exact_gradient_vector_density","operation":"graddeep-negative-direction","semantic_template":"graddeep-negative-direction|{\"gradient\":[[7,1],[-7,1]]}"}},"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-r06-0033","batch_id":"batch-20261015-advance-20261119-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-negative-direction；明示資料={\"gradient\":[8,-8]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([8, -8], [-8, 8], -128, 128, 是, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)","explanation_zh":"Wave6第十三層核對=([8, -8], [-8, 8], -128, 128, 是, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod50-graddeep-negative-direction","seed":202611196033,"parameters":{"gradient":[8,-8],"domain":"Exact_gradient_vector_density","operation":"graddeep-negative-direction","semantic_template":"graddeep-negative-direction|{\"gradient\":[[8,1],[-8,1]]}"}},"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-r06-0034","batch_id":"batch-20261015-advance-20261119-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-negative-direction；明示資料={\"gradient\":[9,-9]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([9, -9], [-9, 9], -162, 162, 是, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)","explanation_zh":"Wave6第十三層核對=([9, -9], [-9, 9], -162, 162, 是, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod50-graddeep-negative-direction","seed":202611196034,"parameters":{"gradient":[9,-9],"domain":"Exact_gradient_vector_density","operation":"graddeep-negative-direction","semantic_template":"graddeep-negative-direction|{\"gradient\":[[9,1],[-9,1]]}"}},"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-r06-0035","batch_id":"batch-20261015-advance-20261119-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-negative-direction；明示資料={\"gradient\":[10,-10]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([10, -10], [-10, 10], -200, 200, 是, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)","explanation_zh":"Wave6第十三層核對=([10, -10], [-10, 10], -200, 200, 是, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod50-graddeep-negative-direction","seed":202611196035,"parameters":{"gradient":[10,-10],"domain":"Exact_gradient_vector_density","operation":"graddeep-negative-direction","semantic_template":"graddeep-negative-direction|{\"gradient\":[[10,1],[-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-r06-0036","batch_id":"batch-20261015-advance-20261119-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-negative-direction；明示資料={\"gradient\":[11,-11]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([11, -11], [-11, 11], -242, 242, 是, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)","explanation_zh":"Wave6第十三層核對=([11, -11], [-11, 11], -242, 242, 是, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod50-graddeep-negative-direction","seed":202611196036,"parameters":{"gradient":[11,-11],"domain":"Exact_gradient_vector_density","operation":"graddeep-negative-direction","semantic_template":"graddeep-negative-direction|{\"gradient\":[[11,1],[-11,1]]}"}},"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-r06-0037","batch_id":"batch-20261015-advance-20261119-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-negative-direction；明示資料={\"gradient\":[12,-12]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([12, -12], [-12, 12], -288, 288, 是, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)","explanation_zh":"Wave6第十三層核對=([12, -12], [-12, 12], -288, 288, 是, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod50-graddeep-negative-direction","seed":202611196037,"parameters":{"gradient":[12,-12],"domain":"Exact_gradient_vector_density","operation":"graddeep-negative-direction","semantic_template":"graddeep-negative-direction|{\"gradient\":[[12,1],[-12,1]]}"}},"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-r06-0038","batch_id":"batch-20261015-advance-20261119-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-negative-direction；明示資料={\"gradient\":[13,-13]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([13, -13], [-13, 13], -338, 338, 是, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)","explanation_zh":"Wave6第十三層核對=([13, -13], [-13, 13], -338, 338, 是, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod50-graddeep-negative-direction","seed":202611196038,"parameters":{"gradient":[13,-13],"domain":"Exact_gradient_vector_density","operation":"graddeep-negative-direction","semantic_template":"graddeep-negative-direction|{\"gradient\":[[13,1],[-13,1]]}"}},"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-r06-0039","batch_id":"batch-20261015-advance-20261119-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-negative-direction；明示資料={\"gradient\":[14,-14]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([14, -14], [-14, 14], -392, 392, 是, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)","explanation_zh":"Wave6第十三層核對=([14, -14], [-14, 14], -392, 392, 是, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod50-graddeep-negative-direction","seed":202611196039,"parameters":{"gradient":[14,-14],"domain":"Exact_gradient_vector_density","operation":"graddeep-negative-direction","semantic_template":"graddeep-negative-direction|{\"gradient\":[[14,1],[-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-r06-0040","batch_id":"batch-20261015-advance-20261119-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-negative-direction；明示資料={\"gradient\":[15,-15]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([15, -15], [-15, 15], -450, 450, 是, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)","explanation_zh":"Wave6第十三層核對=([15, -15], [-15, 15], -450, 450, 是, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod50-graddeep-negative-direction","seed":202611196040,"parameters":{"gradient":[15,-15],"domain":"Exact_gradient_vector_density","operation":"graddeep-negative-direction","semantic_template":"graddeep-negative-direction|{\"gradient\":[[15,1],[-15,1]]}"}},"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-r06-0041","batch_id":"batch-20261015-advance-20261119-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-negative-direction；明示資料={\"gradient\":[16,-16]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([16, -16], [-16, 16], -512, 512, 是, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)","explanation_zh":"Wave6第十三層核對=([16, -16], [-16, 16], -512, 512, 是, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod50-graddeep-negative-direction","seed":202611196041,"parameters":{"gradient":[16,-16],"domain":"Exact_gradient_vector_density","operation":"graddeep-negative-direction","semantic_template":"graddeep-negative-direction|{\"gradient\":[[16,1],[-16,1]]}"}},"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-r06-0042","batch_id":"batch-20261015-advance-20261119-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-negative-direction；明示資料={\"gradient\":[17,-17]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([17, -17], [-17, 17], -578, 578, 是, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)","explanation_zh":"Wave6第十三層核對=([17, -17], [-17, 17], -578, 578, 是, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod50-graddeep-negative-direction","seed":202611196042,"parameters":{"gradient":[17,-17],"domain":"Exact_gradient_vector_density","operation":"graddeep-negative-direction","semantic_template":"graddeep-negative-direction|{\"gradient\":[[17,1],[-17,1]]}"}},"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-r06-0043","batch_id":"batch-20261015-advance-20261119-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-negative-direction；明示資料={\"gradient\":[18,-18]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([18, -18], [-18, 18], -648, 648, 是, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)","explanation_zh":"Wave6第十三層核對=([18, -18], [-18, 18], -648, 648, 是, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod50-graddeep-negative-direction","seed":202611196043,"parameters":{"gradient":[18,-18],"domain":"Exact_gradient_vector_density","operation":"graddeep-negative-direction","semantic_template":"graddeep-negative-direction|{\"gradient\":[[18,1],[-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-r06-0044","batch_id":"batch-20261015-advance-20261119-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-negative-direction；明示資料={\"gradient\":[19,-19]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([19, -19], [-19, 19], -722, 722, 是, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)","explanation_zh":"Wave6第十三層核對=([19, -19], [-19, 19], -722, 722, 是, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod50-graddeep-negative-direction","seed":202611196044,"parameters":{"gradient":[19,-19],"domain":"Exact_gradient_vector_density","operation":"graddeep-negative-direction","semantic_template":"graddeep-negative-direction|{\"gradient\":[[19,1],[-19,1]]}"}},"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-r06-0045","batch_id":"batch-20261015-advance-20261119-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-negative-direction；明示資料={\"gradient\":[20,-20]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([20, -20], [-20, 20], -800, 800, 是, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)","explanation_zh":"Wave6第十三層核對=([20, -20], [-20, 20], -800, 800, 是, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod50-graddeep-negative-direction","seed":202611196045,"parameters":{"gradient":[20,-20],"domain":"Exact_gradient_vector_density","operation":"graddeep-negative-direction","semantic_template":"graddeep-negative-direction|{\"gradient\":[[20,1],[-20,1]]}"}},"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-r06-0046","batch_id":"batch-20261015-advance-20261119-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-negative-direction；明示資料={\"gradient\":[21,-21]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([21, -21], [-21, 21], -882, 882, 是, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)","explanation_zh":"Wave6第十三層核對=([21, -21], [-21, 21], -882, 882, 是, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod50-graddeep-negative-direction","seed":202611196046,"parameters":{"gradient":[21,-21],"domain":"Exact_gradient_vector_density","operation":"graddeep-negative-direction","semantic_template":"graddeep-negative-direction|{\"gradient\":[[21,1],[-21,1]]}"}},"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-r06-0047","batch_id":"batch-20261015-advance-20261119-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-negative-direction；明示資料={\"gradient\":[22,-22]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([22, -22], [-22, 22], -968, 968, 是, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)","explanation_zh":"Wave6第十三層核對=([22, -22], [-22, 22], -968, 968, 是, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod50-graddeep-negative-direction","seed":202611196047,"parameters":{"gradient":[22,-22],"domain":"Exact_gradient_vector_density","operation":"graddeep-negative-direction","semantic_template":"graddeep-negative-direction|{\"gradient\":[[22,1],[-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-r06-0048","batch_id":"batch-20261015-advance-20261119-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-negative-direction；明示資料={\"gradient\":[23,-23]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([23, -23], [-23, 23], -1058, 1058, 是, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)","explanation_zh":"Wave6第十三層核對=([23, -23], [-23, 23], -1058, 1058, 是, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod50-graddeep-negative-direction","seed":202611196048,"parameters":{"gradient":[23,-23],"domain":"Exact_gradient_vector_density","operation":"graddeep-negative-direction","semantic_template":"graddeep-negative-direction|{\"gradient\":[[23,1],[-23,1]]}"}},"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-r06-0049","batch_id":"batch-20261015-advance-20261119-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-negative-direction；明示資料={\"gradient\":[24,-24]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([24, -24], [-24, 24], -1152, 1152, 是, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)","explanation_zh":"Wave6第十三層核對=([24, -24], [-24, 24], -1152, 1152, 是, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod50-graddeep-negative-direction","seed":202611196049,"parameters":{"gradient":[24,-24],"domain":"Exact_gradient_vector_density","operation":"graddeep-negative-direction","semantic_template":"graddeep-negative-direction|{\"gradient\":[[24,1],[-24,1]]}"}},"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-r06-0050","batch_id":"batch-20261015-advance-20261119-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-negative-direction；明示資料={\"gradient\":[25,-25]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([25, -25], [-25, 25], -1250, 1250, 是, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)","explanation_zh":"Wave6第十三層核對=([25, -25], [-25, 25], -1250, 1250, 是, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod50-graddeep-negative-direction","seed":202611196050,"parameters":{"gradient":[25,-25],"domain":"Exact_gradient_vector_density","operation":"graddeep-negative-direction","semantic_template":"graddeep-negative-direction|{\"gradient\":[[25,1],[-25,1]]}"}},"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-r06-0051","batch_id":"batch-20261015-advance-20261119-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-quadratic-difference；明示資料={\"symmetric_matrix\":[[1,1],[1,2]],\"x\":[0,1],\"y\":[1,0]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([[1, 1], [1, 2]], [0, 1], [1, 0], [-1, 1], [0, 1], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)","explanation_zh":"Wave6第十三層核對=([[1, 1], [1, 2]], [0, 1], [1, 0], [-1, 1], [0, 1], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod50-graddeep-quadratic-difference","seed":202611196051,"parameters":{"symmetric_matrix":[[1,1],[1,2]],"x":[0,1],"y":[1,0],"domain":"Exact_gradient_vector_density","operation":"graddeep-quadratic-difference","semantic_template":"graddeep-quadratic-difference|{\"symmetric_matrix\":[[[1,1],[1,1]],[[1,1],[2,1]]],\"x\":[[0,1],[1,1]],\"y\":[[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-r06-0052","batch_id":"batch-20261015-advance-20261119-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-quadratic-difference；明示資料={\"symmetric_matrix\":[[2,1],[1,3]],\"x\":[1,1],\"y\":[1,-1]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([[2, 1], [1, 3]], [1, 1], [1, -1], [0, 2], [2, 6], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)","explanation_zh":"Wave6第十三層核對=([[2, 1], [1, 3]], [1, 1], [1, -1], [0, 2], [2, 6], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod50-graddeep-quadratic-difference","seed":202611196052,"parameters":{"symmetric_matrix":[[2,1],[1,3]],"x":[1,1],"y":[1,-1],"domain":"Exact_gradient_vector_density","operation":"graddeep-quadratic-difference","semantic_template":"graddeep-quadratic-difference|{\"symmetric_matrix\":[[[2,1],[1,1]],[[1,1],[3,1]]],\"x\":[[1,1],[1,1]],\"y\":[[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-r06-0053","batch_id":"batch-20261015-advance-20261119-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-quadratic-difference；明示資料={\"symmetric_matrix\":[[3,1],[1,4]],\"x\":[2,1],\"y\":[1,-2]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([[3, 1], [1, 4]], [2, 1], [1, -2], [1, 3], [6, 13], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)","explanation_zh":"Wave6第十三層核對=([[3, 1], [1, 4]], [2, 1], [1, -2], [1, 3], [6, 13], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod50-graddeep-quadratic-difference","seed":202611196053,"parameters":{"symmetric_matrix":[[3,1],[1,4]],"x":[2,1],"y":[1,-2],"domain":"Exact_gradient_vector_density","operation":"graddeep-quadratic-difference","semantic_template":"graddeep-quadratic-difference|{\"symmetric_matrix\":[[[3,1],[1,1]],[[1,1],[4,1]]],\"x\":[[2,1],[1,1]],\"y\":[[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-r06-0054","batch_id":"batch-20261015-advance-20261119-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-quadratic-difference；明示資料={\"symmetric_matrix\":[[4,1],[1,5]],\"x\":[3,1],\"y\":[1,-3]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([[4, 1], [1, 5]], [3, 1], [1, -3], [2, 4], [12, 22], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)","explanation_zh":"Wave6第十三層核對=([[4, 1], [1, 5]], [3, 1], [1, -3], [2, 4], [12, 22], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod50-graddeep-quadratic-difference","seed":202611196054,"parameters":{"symmetric_matrix":[[4,1],[1,5]],"x":[3,1],"y":[1,-3],"domain":"Exact_gradient_vector_density","operation":"graddeep-quadratic-difference","semantic_template":"graddeep-quadratic-difference|{\"symmetric_matrix\":[[[4,1],[1,1]],[[1,1],[5,1]]],\"x\":[[3,1],[1,1]],\"y\":[[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-r06-0055","batch_id":"batch-20261015-advance-20261119-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-quadratic-difference；明示資料={\"symmetric_matrix\":[[5,1],[1,6]],\"x\":[4,1],\"y\":[1,-4]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([[5, 1], [1, 6]], [4, 1], [1, -4], [3, 5], [20, 33], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)","explanation_zh":"Wave6第十三層核對=([[5, 1], [1, 6]], [4, 1], [1, -4], [3, 5], [20, 33], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod50-graddeep-quadratic-difference","seed":202611196055,"parameters":{"symmetric_matrix":[[5,1],[1,6]],"x":[4,1],"y":[1,-4],"domain":"Exact_gradient_vector_density","operation":"graddeep-quadratic-difference","semantic_template":"graddeep-quadratic-difference|{\"symmetric_matrix\":[[[5,1],[1,1]],[[1,1],[6,1]]],\"x\":[[4,1],[1,1]],\"y\":[[1,1],[-4,1]]}"}},"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-r06-0056","batch_id":"batch-20261015-advance-20261119-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-quadratic-difference；明示資料={\"symmetric_matrix\":[[6,1],[1,2]],\"x\":[5,1],\"y\":[1,-5]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([[6, 1], [1, 2]], [5, 1], [1, -5], [4, 6], [30, 16], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)","explanation_zh":"Wave6第十三層核對=([[6, 1], [1, 2]], [5, 1], [1, -5], [4, 6], [30, 16], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod50-graddeep-quadratic-difference","seed":202611196056,"parameters":{"symmetric_matrix":[[6,1],[1,2]],"x":[5,1],"y":[1,-5],"domain":"Exact_gradient_vector_density","operation":"graddeep-quadratic-difference","semantic_template":"graddeep-quadratic-difference|{\"symmetric_matrix\":[[[6,1],[1,1]],[[1,1],[2,1]]],\"x\":[[5,1],[1,1]],\"y\":[[1,1],[-5,1]]}"}},"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-r06-0057","batch_id":"batch-20261015-advance-20261119-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-quadratic-difference；明示資料={\"symmetric_matrix\":[[7,1],[1,3]],\"x\":[6,1],\"y\":[1,-6]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([[7, 1], [1, 3]], [6, 1], [1, -6], [5, 7], [42, 26], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)","explanation_zh":"Wave6第十三層核對=([[7, 1], [1, 3]], [6, 1], [1, -6], [5, 7], [42, 26], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod50-graddeep-quadratic-difference","seed":202611196057,"parameters":{"symmetric_matrix":[[7,1],[1,3]],"x":[6,1],"y":[1,-6],"domain":"Exact_gradient_vector_density","operation":"graddeep-quadratic-difference","semantic_template":"graddeep-quadratic-difference|{\"symmetric_matrix\":[[[7,1],[1,1]],[[1,1],[3,1]]],\"x\":[[6,1],[1,1]],\"y\":[[1,1],[-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-r06-0058","batch_id":"batch-20261015-advance-20261119-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-quadratic-difference；明示資料={\"symmetric_matrix\":[[8,1],[1,4]],\"x\":[7,1],\"y\":[1,-7]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([[8, 1], [1, 4]], [7, 1], [1, -7], [6, 8], [56, 38], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)","explanation_zh":"Wave6第十三層核對=([[8, 1], [1, 4]], [7, 1], [1, -7], [6, 8], [56, 38], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod50-graddeep-quadratic-difference","seed":202611196058,"parameters":{"symmetric_matrix":[[8,1],[1,4]],"x":[7,1],"y":[1,-7],"domain":"Exact_gradient_vector_density","operation":"graddeep-quadratic-difference","semantic_template":"graddeep-quadratic-difference|{\"symmetric_matrix\":[[[8,1],[1,1]],[[1,1],[4,1]]],\"x\":[[7,1],[1,1]],\"y\":[[1,1],[-7,1]]}"}},"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-r06-0059","batch_id":"batch-20261015-advance-20261119-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-quadratic-difference；明示資料={\"symmetric_matrix\":[[9,1],[1,5]],\"x\":[8,1],\"y\":[1,-8]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([[9, 1], [1, 5]], [8, 1], [1, -8], [7, 9], [72, 52], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)","explanation_zh":"Wave6第十三層核對=([[9, 1], [1, 5]], [8, 1], [1, -8], [7, 9], [72, 52], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod50-graddeep-quadratic-difference","seed":202611196059,"parameters":{"symmetric_matrix":[[9,1],[1,5]],"x":[8,1],"y":[1,-8],"domain":"Exact_gradient_vector_density","operation":"graddeep-quadratic-difference","semantic_template":"graddeep-quadratic-difference|{\"symmetric_matrix\":[[[9,1],[1,1]],[[1,1],[5,1]]],\"x\":[[8,1],[1,1]],\"y\":[[1,1],[-8,1]]}"}},"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-r06-0060","batch_id":"batch-20261015-advance-20261119-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-quadratic-difference；明示資料={\"symmetric_matrix\":[[10,1],[1,6]],\"x\":[9,1],\"y\":[1,-9]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([[10, 1], [1, 6]], [9, 1], [1, -9], [8, 10], [90, 68], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)","explanation_zh":"Wave6第十三層核對=([[10, 1], [1, 6]], [9, 1], [1, -9], [8, 10], [90, 68], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod50-graddeep-quadratic-difference","seed":202611196060,"parameters":{"symmetric_matrix":[[10,1],[1,6]],"x":[9,1],"y":[1,-9],"domain":"Exact_gradient_vector_density","operation":"graddeep-quadratic-difference","semantic_template":"graddeep-quadratic-difference|{\"symmetric_matrix\":[[[10,1],[1,1]],[[1,1],[6,1]]],\"x\":[[9,1],[1,1]],\"y\":[[1,1],[-9,1]]}"}},"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-r06-0061","batch_id":"batch-20261015-advance-20261119-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-quadratic-difference；明示資料={\"symmetric_matrix\":[[11,1],[1,2]],\"x\":[10,1],\"y\":[1,-10]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([[11, 1], [1, 2]], [10, 1], [1, -10], [9, 11], [110, 31], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)","explanation_zh":"Wave6第十三層核對=([[11, 1], [1, 2]], [10, 1], [1, -10], [9, 11], [110, 31], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod50-graddeep-quadratic-difference","seed":202611196061,"parameters":{"symmetric_matrix":[[11,1],[1,2]],"x":[10,1],"y":[1,-10],"domain":"Exact_gradient_vector_density","operation":"graddeep-quadratic-difference","semantic_template":"graddeep-quadratic-difference|{\"symmetric_matrix\":[[[11,1],[1,1]],[[1,1],[2,1]]],\"x\":[[10,1],[1,1]],\"y\":[[1,1],[-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-r06-0062","batch_id":"batch-20261015-advance-20261119-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-quadratic-difference；明示資料={\"symmetric_matrix\":[[12,1],[1,3]],\"x\":[11,1],\"y\":[1,-11]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([[12, 1], [1, 3]], [11, 1], [1, -11], [10, 12], [132, 46], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)","explanation_zh":"Wave6第十三層核對=([[12, 1], [1, 3]], [11, 1], [1, -11], [10, 12], [132, 46], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod50-graddeep-quadratic-difference","seed":202611196062,"parameters":{"symmetric_matrix":[[12,1],[1,3]],"x":[11,1],"y":[1,-11],"domain":"Exact_gradient_vector_density","operation":"graddeep-quadratic-difference","semantic_template":"graddeep-quadratic-difference|{\"symmetric_matrix\":[[[12,1],[1,1]],[[1,1],[3,1]]],\"x\":[[11,1],[1,1]],\"y\":[[1,1],[-11,1]]}"}},"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-r06-0063","batch_id":"batch-20261015-advance-20261119-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-quadratic-difference；明示資料={\"symmetric_matrix\":[[13,1],[1,4]],\"x\":[12,1],\"y\":[1,-12]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([[13, 1], [1, 4]], [12, 1], [1, -12], [11, 13], [156, 63], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)","explanation_zh":"Wave6第十三層核對=([[13, 1], [1, 4]], [12, 1], [1, -12], [11, 13], [156, 63], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod50-graddeep-quadratic-difference","seed":202611196063,"parameters":{"symmetric_matrix":[[13,1],[1,4]],"x":[12,1],"y":[1,-12],"domain":"Exact_gradient_vector_density","operation":"graddeep-quadratic-difference","semantic_template":"graddeep-quadratic-difference|{\"symmetric_matrix\":[[[13,1],[1,1]],[[1,1],[4,1]]],\"x\":[[12,1],[1,1]],\"y\":[[1,1],[-12,1]]}"}},"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-r06-0064","batch_id":"batch-20261015-advance-20261119-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-quadratic-difference；明示資料={\"symmetric_matrix\":[[14,1],[1,5]],\"x\":[13,1],\"y\":[1,-13]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([[14, 1], [1, 5]], [13, 1], [1, -13], [12, 14], [182, 82], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)","explanation_zh":"Wave6第十三層核對=([[14, 1], [1, 5]], [13, 1], [1, -13], [12, 14], [182, 82], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod50-graddeep-quadratic-difference","seed":202611196064,"parameters":{"symmetric_matrix":[[14,1],[1,5]],"x":[13,1],"y":[1,-13],"domain":"Exact_gradient_vector_density","operation":"graddeep-quadratic-difference","semantic_template":"graddeep-quadratic-difference|{\"symmetric_matrix\":[[[14,1],[1,1]],[[1,1],[5,1]]],\"x\":[[13,1],[1,1]],\"y\":[[1,1],[-13,1]]}"}},"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-r06-0065","batch_id":"batch-20261015-advance-20261119-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-quadratic-difference；明示資料={\"symmetric_matrix\":[[15,1],[1,6]],\"x\":[14,1],\"y\":[1,-14]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([[15, 1], [1, 6]], [14, 1], [1, -14], [13, 15], [210, 103], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)","explanation_zh":"Wave6第十三層核對=([[15, 1], [1, 6]], [14, 1], [1, -14], [13, 15], [210, 103], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod50-graddeep-quadratic-difference","seed":202611196065,"parameters":{"symmetric_matrix":[[15,1],[1,6]],"x":[14,1],"y":[1,-14],"domain":"Exact_gradient_vector_density","operation":"graddeep-quadratic-difference","semantic_template":"graddeep-quadratic-difference|{\"symmetric_matrix\":[[[15,1],[1,1]],[[1,1],[6,1]]],\"x\":[[14,1],[1,1]],\"y\":[[1,1],[-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-r06-0066","batch_id":"batch-20261015-advance-20261119-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-quadratic-difference；明示資料={\"symmetric_matrix\":[[16,1],[1,2]],\"x\":[15,1],\"y\":[1,-15]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([[16, 1], [1, 2]], [15, 1], [1, -15], [14, 16], [240, 46], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)","explanation_zh":"Wave6第十三層核對=([[16, 1], [1, 2]], [15, 1], [1, -15], [14, 16], [240, 46], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod50-graddeep-quadratic-difference","seed":202611196066,"parameters":{"symmetric_matrix":[[16,1],[1,2]],"x":[15,1],"y":[1,-15],"domain":"Exact_gradient_vector_density","operation":"graddeep-quadratic-difference","semantic_template":"graddeep-quadratic-difference|{\"symmetric_matrix\":[[[16,1],[1,1]],[[1,1],[2,1]]],\"x\":[[15,1],[1,1]],\"y\":[[1,1],[-15,1]]}"}},"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-r06-0067","batch_id":"batch-20261015-advance-20261119-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-quadratic-difference；明示資料={\"symmetric_matrix\":[[17,1],[1,3]],\"x\":[16,1],\"y\":[1,-16]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([[17, 1], [1, 3]], [16, 1], [1, -16], [15, 17], [272, 66], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)","explanation_zh":"Wave6第十三層核對=([[17, 1], [1, 3]], [16, 1], [1, -16], [15, 17], [272, 66], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod50-graddeep-quadratic-difference","seed":202611196067,"parameters":{"symmetric_matrix":[[17,1],[1,3]],"x":[16,1],"y":[1,-16],"domain":"Exact_gradient_vector_density","operation":"graddeep-quadratic-difference","semantic_template":"graddeep-quadratic-difference|{\"symmetric_matrix\":[[[17,1],[1,1]],[[1,1],[3,1]]],\"x\":[[16,1],[1,1]],\"y\":[[1,1],[-16,1]]}"}},"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-r06-0068","batch_id":"batch-20261015-advance-20261119-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-quadratic-difference；明示資料={\"symmetric_matrix\":[[18,1],[1,4]],\"x\":[17,1],\"y\":[1,-17]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([[18, 1], [1, 4]], [17, 1], [1, -17], [16, 18], [306, 88], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)","explanation_zh":"Wave6第十三層核對=([[18, 1], [1, 4]], [17, 1], [1, -17], [16, 18], [306, 88], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod50-graddeep-quadratic-difference","seed":202611196068,"parameters":{"symmetric_matrix":[[18,1],[1,4]],"x":[17,1],"y":[1,-17],"domain":"Exact_gradient_vector_density","operation":"graddeep-quadratic-difference","semantic_template":"graddeep-quadratic-difference|{\"symmetric_matrix\":[[[18,1],[1,1]],[[1,1],[4,1]]],\"x\":[[17,1],[1,1]],\"y\":[[1,1],[-17,1]]}"}},"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-r06-0069","batch_id":"batch-20261015-advance-20261119-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-quadratic-difference；明示資料={\"symmetric_matrix\":[[19,1],[1,5]],\"x\":[18,1],\"y\":[1,-18]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([[19, 1], [1, 5]], [18, 1], [1, -18], [17, 19], [342, 112], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)","explanation_zh":"Wave6第十三層核對=([[19, 1], [1, 5]], [18, 1], [1, -18], [17, 19], [342, 112], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod50-graddeep-quadratic-difference","seed":202611196069,"parameters":{"symmetric_matrix":[[19,1],[1,5]],"x":[18,1],"y":[1,-18],"domain":"Exact_gradient_vector_density","operation":"graddeep-quadratic-difference","semantic_template":"graddeep-quadratic-difference|{\"symmetric_matrix\":[[[19,1],[1,1]],[[1,1],[5,1]]],\"x\":[[18,1],[1,1]],\"y\":[[1,1],[-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-r06-0070","batch_id":"batch-20261015-advance-20261119-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-quadratic-difference；明示資料={\"symmetric_matrix\":[[20,1],[1,6]],\"x\":[19,1],\"y\":[1,-19]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([[20, 1], [1, 6]], [19, 1], [1, -19], [18, 20], [380, 138], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)","explanation_zh":"Wave6第十三層核對=([[20, 1], [1, 6]], [19, 1], [1, -19], [18, 20], [380, 138], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod50-graddeep-quadratic-difference","seed":202611196070,"parameters":{"symmetric_matrix":[[20,1],[1,6]],"x":[19,1],"y":[1,-19],"domain":"Exact_gradient_vector_density","operation":"graddeep-quadratic-difference","semantic_template":"graddeep-quadratic-difference|{\"symmetric_matrix\":[[[20,1],[1,1]],[[1,1],[6,1]]],\"x\":[[19,1],[1,1]],\"y\":[[1,1],[-19,1]]}"}},"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-r06-0071","batch_id":"batch-20261015-advance-20261119-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-quadratic-difference；明示資料={\"symmetric_matrix\":[[21,1],[1,2]],\"x\":[20,1],\"y\":[1,-20]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([[21, 1], [1, 2]], [20, 1], [1, -20], [19, 21], [420, 61], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)","explanation_zh":"Wave6第十三層核對=([[21, 1], [1, 2]], [20, 1], [1, -20], [19, 21], [420, 61], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod50-graddeep-quadratic-difference","seed":202611196071,"parameters":{"symmetric_matrix":[[21,1],[1,2]],"x":[20,1],"y":[1,-20],"domain":"Exact_gradient_vector_density","operation":"graddeep-quadratic-difference","semantic_template":"graddeep-quadratic-difference|{\"symmetric_matrix\":[[[21,1],[1,1]],[[1,1],[2,1]]],\"x\":[[20,1],[1,1]],\"y\":[[1,1],[-20,1]]}"}},"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-r06-0072","batch_id":"batch-20261015-advance-20261119-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-quadratic-difference；明示資料={\"symmetric_matrix\":[[22,1],[1,3]],\"x\":[21,1],\"y\":[1,-21]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([[22, 1], [1, 3]], [21, 1], [1, -21], [20, 22], [462, 86], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)","explanation_zh":"Wave6第十三層核對=([[22, 1], [1, 3]], [21, 1], [1, -21], [20, 22], [462, 86], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod50-graddeep-quadratic-difference","seed":202611196072,"parameters":{"symmetric_matrix":[[22,1],[1,3]],"x":[21,1],"y":[1,-21],"domain":"Exact_gradient_vector_density","operation":"graddeep-quadratic-difference","semantic_template":"graddeep-quadratic-difference|{\"symmetric_matrix\":[[[22,1],[1,1]],[[1,1],[3,1]]],\"x\":[[21,1],[1,1]],\"y\":[[1,1],[-21,1]]}"}},"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-r06-0073","batch_id":"batch-20261015-advance-20261119-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-quadratic-difference；明示資料={\"symmetric_matrix\":[[23,1],[1,4]],\"x\":[22,1],\"y\":[1,-22]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([[23, 1], [1, 4]], [22, 1], [1, -22], [21, 23], [506, 113], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)","explanation_zh":"Wave6第十三層核對=([[23, 1], [1, 4]], [22, 1], [1, -22], [21, 23], [506, 113], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod50-graddeep-quadratic-difference","seed":202611196073,"parameters":{"symmetric_matrix":[[23,1],[1,4]],"x":[22,1],"y":[1,-22],"domain":"Exact_gradient_vector_density","operation":"graddeep-quadratic-difference","semantic_template":"graddeep-quadratic-difference|{\"symmetric_matrix\":[[[23,1],[1,1]],[[1,1],[4,1]]],\"x\":[[22,1],[1,1]],\"y\":[[1,1],[-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-r06-0074","batch_id":"batch-20261015-advance-20261119-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-quadratic-difference；明示資料={\"symmetric_matrix\":[[24,1],[1,5]],\"x\":[23,1],\"y\":[1,-23]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([[24, 1], [1, 5]], [23, 1], [1, -23], [22, 24], [552, 142], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)","explanation_zh":"Wave6第十三層核對=([[24, 1], [1, 5]], [23, 1], [1, -23], [22, 24], [552, 142], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod50-graddeep-quadratic-difference","seed":202611196074,"parameters":{"symmetric_matrix":[[24,1],[1,5]],"x":[23,1],"y":[1,-23],"domain":"Exact_gradient_vector_density","operation":"graddeep-quadratic-difference","semantic_template":"graddeep-quadratic-difference|{\"symmetric_matrix\":[[[24,1],[1,1]],[[1,1],[5,1]]],\"x\":[[23,1],[1,1]],\"y\":[[1,1],[-23,1]]}"}},"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-r06-0075","batch_id":"batch-20261015-advance-20261119-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-quadratic-difference；明示資料={\"symmetric_matrix\":[[25,1],[1,6]],\"x\":[24,1],\"y\":[1,-24]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([[25, 1], [1, 6]], [24, 1], [1, -24], [23, 25], [600, 173], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)","explanation_zh":"Wave6第十三層核對=([[25, 1], [1, 6]], [24, 1], [1, -24], [23, 25], [600, 173], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod50-graddeep-quadratic-difference","seed":202611196075,"parameters":{"symmetric_matrix":[[25,1],[1,6]],"x":[24,1],"y":[1,-24],"domain":"Exact_gradient_vector_density","operation":"graddeep-quadratic-difference","semantic_template":"graddeep-quadratic-difference|{\"symmetric_matrix\":[[[25,1],[1,1]],[[1,1],[6,1]]],\"x\":[[24,1],[1,1]],\"y\":[[1,1],[-24,1]]}"}},"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-r06-0076","batch_id":"batch-20261015-advance-20261119-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-boundary；明示資料={\"mode\":\"zero_gradient\",\"model\":{\"dimension\":1}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(zero_gradient, 1, [0], 是, 否, zero_gradient_only_certifies_stationarity)","explanation_zh":"Wave6第十三層核對=(zero_gradient, 1, [0], 是, 否, zero_gradient_only_certifies_stationarity)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod50-graddeep-boundary","seed":202611196076,"parameters":{"mode":"zero_gradient","model":{"dimension":1},"domain":"Exact_gradient_vector_density","operation":"graddeep-boundary","semantic_template":"graddeep-boundary|{\"mode\":\"zero_gradient\",\"model\":{\"dimension\":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-r06-0077","batch_id":"batch-20261015-advance-20261119-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-boundary；明示資料={\"mode\":\"positive_gradient_direction\",\"model\":{\"gradient\":[2,-1]}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(positive_gradient_direction, [2, -1], [2, -1], 5, 是, 否, positive_gradient_is_first_order_ascent_not_descent)","explanation_zh":"Wave6第十三層核對=(positive_gradient_direction, [2, -1], [2, -1], 5, 是, 否, positive_gradient_is_first_order_ascent_not_descent)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod50-graddeep-boundary","seed":202611196077,"parameters":{"mode":"positive_gradient_direction","model":{"gradient":[2,-1]},"domain":"Exact_gradient_vector_density","operation":"graddeep-boundary","semantic_template":"graddeep-boundary|{\"mode\":\"positive_gradient_direction\",\"model\":{\"gradient\":[[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-r06-0078","batch_id":"batch-20261015-advance-20261119-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-boundary；明示資料={\"mode\":\"non_euclidean_metric\",\"model\":{\"coordinate_gradient\":[3,1],\"metric\":[[2,0],[0,3]]}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(non_euclidean_metric, [3, 1], [[2, 0], [0, 3]], [3/2, 1/3], 否, coordinate_partials_become_a_gradient_vector_only_after_the_metric_identification)","explanation_zh":"Wave6第十三層核對=(non_euclidean_metric, [3, 1], [[2, 0], [0, 3]], [3/2, 1/3], 否, coordinate_partials_become_a_gradient_vector_only_after_the_metric_identification)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod50-graddeep-boundary","seed":202611196078,"parameters":{"mode":"non_euclidean_metric","model":{"coordinate_gradient":[3,1],"metric":[[2,0],[0,3]]},"domain":"Exact_gradient_vector_density","operation":"graddeep-boundary","semantic_template":"graddeep-boundary|{\"mode\":\"non_euclidean_metric\",\"model\":{\"coordinate_gradient\":[[3,1],[1,1]],\"metric\":[[[2,1],[0,1]],[[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-r06-0079","batch_id":"batch-20261015-advance-20261119-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-boundary；明示資料={\"mode\":\"finite_difference_confusion\",\"model\":{\"difference_quotient\":4,\"step\":[1,5]}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(finite_difference_confusion, 4, 1/5, 否, one_finite_difference_quotient_is_not_a_derivative_or_gradient_certificate)","explanation_zh":"Wave6第十三層核對=(finite_difference_confusion, 4, 1/5, 否, one_finite_difference_quotient_is_not_a_derivative_or_gradient_certificate)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod50-graddeep-boundary","seed":202611196079,"parameters":{"mode":"finite_difference_confusion","model":{"difference_quotient":4,"step":[1,5]},"domain":"Exact_gradient_vector_density","operation":"graddeep-boundary","semantic_template":"graddeep-boundary|{\"mode\":\"finite_difference_confusion\",\"model\":{\"difference_quotient\":[4,1],\"step\":[1,5]}}"}},"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-r06-0080","batch_id":"batch-20261015-advance-20261119-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-boundary；明示資料={\"mode\":\"finite_difference_confusion\",\"model\":{\"difference_quotient\":5,\"step\":[1,6]}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(finite_difference_confusion, 5, 1/6, 否, one_finite_difference_quotient_is_not_a_derivative_or_gradient_certificate)","explanation_zh":"Wave6第十三層核對=(finite_difference_confusion, 5, 1/6, 否, one_finite_difference_quotient_is_not_a_derivative_or_gradient_certificate)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod50-graddeep-boundary","seed":202611196080,"parameters":{"mode":"finite_difference_confusion","model":{"difference_quotient":5,"step":[1,6]},"domain":"Exact_gradient_vector_density","operation":"graddeep-boundary","semantic_template":"graddeep-boundary|{\"mode\":\"finite_difference_confusion\",\"model\":{\"difference_quotient\":[5,1],\"step\":[1,6]}}"}},"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-r06-0081","batch_id":"batch-20261015-advance-20261119-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-boundary；明示資料={\"mode\":\"positive_gradient_direction\",\"model\":{\"gradient\":[6,-1]}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(positive_gradient_direction, [6, -1], [6, -1], 37, 是, 否, positive_gradient_is_first_order_ascent_not_descent)","explanation_zh":"Wave6第十三層核對=(positive_gradient_direction, [6, -1], [6, -1], 37, 是, 否, positive_gradient_is_first_order_ascent_not_descent)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod50-graddeep-boundary","seed":202611196081,"parameters":{"mode":"positive_gradient_direction","model":{"gradient":[6,-1]},"domain":"Exact_gradient_vector_density","operation":"graddeep-boundary","semantic_template":"graddeep-boundary|{\"mode\":\"positive_gradient_direction\",\"model\":{\"gradient\":[[6,1],[-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-r06-0082","batch_id":"batch-20261015-advance-20261119-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-boundary；明示資料={\"mode\":\"non_euclidean_metric\",\"model\":{\"coordinate_gradient\":[7,1],\"metric\":[[2,0],[0,3]]}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(non_euclidean_metric, [7, 1], [[2, 0], [0, 3]], [7/2, 1/3], 否, coordinate_partials_become_a_gradient_vector_only_after_the_metric_identification)","explanation_zh":"Wave6第十三層核對=(non_euclidean_metric, [7, 1], [[2, 0], [0, 3]], [7/2, 1/3], 否, coordinate_partials_become_a_gradient_vector_only_after_the_metric_identification)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod50-graddeep-boundary","seed":202611196082,"parameters":{"mode":"non_euclidean_metric","model":{"coordinate_gradient":[7,1],"metric":[[2,0],[0,3]]},"domain":"Exact_gradient_vector_density","operation":"graddeep-boundary","semantic_template":"graddeep-boundary|{\"mode\":\"non_euclidean_metric\",\"model\":{\"coordinate_gradient\":[[7,1],[1,1]],\"metric\":[[[2,1],[0,1]],[[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-r06-0083","batch_id":"batch-20261015-advance-20261119-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-boundary；明示資料={\"mode\":\"finite_difference_confusion\",\"model\":{\"difference_quotient\":8,\"step\":[1,9]}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(finite_difference_confusion, 8, 1/9, 否, one_finite_difference_quotient_is_not_a_derivative_or_gradient_certificate)","explanation_zh":"Wave6第十三層核對=(finite_difference_confusion, 8, 1/9, 否, one_finite_difference_quotient_is_not_a_derivative_or_gradient_certificate)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod50-graddeep-boundary","seed":202611196083,"parameters":{"mode":"finite_difference_confusion","model":{"difference_quotient":8,"step":[1,9]},"domain":"Exact_gradient_vector_density","operation":"graddeep-boundary","semantic_template":"graddeep-boundary|{\"mode\":\"finite_difference_confusion\",\"model\":{\"difference_quotient\":[8,1],\"step\":[1,9]}}"}},"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-r06-0084","batch_id":"batch-20261015-advance-20261119-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-boundary；明示資料={\"mode\":\"finite_difference_confusion\",\"model\":{\"difference_quotient\":9,\"step\":[1,10]}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(finite_difference_confusion, 9, 1/10, 否, one_finite_difference_quotient_is_not_a_derivative_or_gradient_certificate)","explanation_zh":"Wave6第十三層核對=(finite_difference_confusion, 9, 1/10, 否, one_finite_difference_quotient_is_not_a_derivative_or_gradient_certificate)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod50-graddeep-boundary","seed":202611196084,"parameters":{"mode":"finite_difference_confusion","model":{"difference_quotient":9,"step":[1,10]},"domain":"Exact_gradient_vector_density","operation":"graddeep-boundary","semantic_template":"graddeep-boundary|{\"mode\":\"finite_difference_confusion\",\"model\":{\"difference_quotient\":[9,1],\"step\":[1,10]}}"}},"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-r06-0085","batch_id":"batch-20261015-advance-20261119-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-boundary；明示資料={\"mode\":\"positive_gradient_direction\",\"model\":{\"gradient\":[10,-1]}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(positive_gradient_direction, [10, -1], [10, -1], 101, 是, 否, positive_gradient_is_first_order_ascent_not_descent)","explanation_zh":"Wave6第十三層核對=(positive_gradient_direction, [10, -1], [10, -1], 101, 是, 否, positive_gradient_is_first_order_ascent_not_descent)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod50-graddeep-boundary","seed":202611196085,"parameters":{"mode":"positive_gradient_direction","model":{"gradient":[10,-1]},"domain":"Exact_gradient_vector_density","operation":"graddeep-boundary","semantic_template":"graddeep-boundary|{\"mode\":\"positive_gradient_direction\",\"model\":{\"gradient\":[[10,1],[-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-r06-0086","batch_id":"batch-20261015-advance-20261119-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-boundary；明示資料={\"mode\":\"non_euclidean_metric\",\"model\":{\"coordinate_gradient\":[11,1],\"metric\":[[2,0],[0,3]]}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(non_euclidean_metric, [11, 1], [[2, 0], [0, 3]], [11/2, 1/3], 否, coordinate_partials_become_a_gradient_vector_only_after_the_metric_identification)","explanation_zh":"Wave6第十三層核對=(non_euclidean_metric, [11, 1], [[2, 0], [0, 3]], [11/2, 1/3], 否, coordinate_partials_become_a_gradient_vector_only_after_the_metric_identification)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod50-graddeep-boundary","seed":202611196086,"parameters":{"mode":"non_euclidean_metric","model":{"coordinate_gradient":[11,1],"metric":[[2,0],[0,3]]},"domain":"Exact_gradient_vector_density","operation":"graddeep-boundary","semantic_template":"graddeep-boundary|{\"mode\":\"non_euclidean_metric\",\"model\":{\"coordinate_gradient\":[[11,1],[1,1]],\"metric\":[[[2,1],[0,1]],[[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-r06-0087","batch_id":"batch-20261015-advance-20261119-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-boundary；明示資料={\"mode\":\"finite_difference_confusion\",\"model\":{\"difference_quotient\":12,\"step\":[1,13]}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(finite_difference_confusion, 12, 1/13, 否, one_finite_difference_quotient_is_not_a_derivative_or_gradient_certificate)","explanation_zh":"Wave6第十三層核對=(finite_difference_confusion, 12, 1/13, 否, one_finite_difference_quotient_is_not_a_derivative_or_gradient_certificate)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod50-graddeep-boundary","seed":202611196087,"parameters":{"mode":"finite_difference_confusion","model":{"difference_quotient":12,"step":[1,13]},"domain":"Exact_gradient_vector_density","operation":"graddeep-boundary","semantic_template":"graddeep-boundary|{\"mode\":\"finite_difference_confusion\",\"model\":{\"difference_quotient\":[12,1],\"step\":[1,13]}}"}},"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-r06-0088","batch_id":"batch-20261015-advance-20261119-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-boundary；明示資料={\"mode\":\"finite_difference_confusion\",\"model\":{\"difference_quotient\":13,\"step\":[1,14]}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(finite_difference_confusion, 13, 1/14, 否, one_finite_difference_quotient_is_not_a_derivative_or_gradient_certificate)","explanation_zh":"Wave6第十三層核對=(finite_difference_confusion, 13, 1/14, 否, one_finite_difference_quotient_is_not_a_derivative_or_gradient_certificate)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod50-graddeep-boundary","seed":202611196088,"parameters":{"mode":"finite_difference_confusion","model":{"difference_quotient":13,"step":[1,14]},"domain":"Exact_gradient_vector_density","operation":"graddeep-boundary","semantic_template":"graddeep-boundary|{\"mode\":\"finite_difference_confusion\",\"model\":{\"difference_quotient\":[13,1],\"step\":[1,14]}}"}},"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-r06-0089","batch_id":"batch-20261015-advance-20261119-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-boundary；明示資料={\"mode\":\"positive_gradient_direction\",\"model\":{\"gradient\":[14,-1]}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(positive_gradient_direction, [14, -1], [14, -1], 197, 是, 否, positive_gradient_is_first_order_ascent_not_descent)","explanation_zh":"Wave6第十三層核對=(positive_gradient_direction, [14, -1], [14, -1], 197, 是, 否, positive_gradient_is_first_order_ascent_not_descent)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod50-graddeep-boundary","seed":202611196089,"parameters":{"mode":"positive_gradient_direction","model":{"gradient":[14,-1]},"domain":"Exact_gradient_vector_density","operation":"graddeep-boundary","semantic_template":"graddeep-boundary|{\"mode\":\"positive_gradient_direction\",\"model\":{\"gradient\":[[14,1],[-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-r06-0090","batch_id":"batch-20261015-advance-20261119-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-boundary；明示資料={\"mode\":\"non_euclidean_metric\",\"model\":{\"coordinate_gradient\":[15,1],\"metric\":[[2,0],[0,3]]}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(non_euclidean_metric, [15, 1], [[2, 0], [0, 3]], [15/2, 1/3], 否, coordinate_partials_become_a_gradient_vector_only_after_the_metric_identification)","explanation_zh":"Wave6第十三層核對=(non_euclidean_metric, [15, 1], [[2, 0], [0, 3]], [15/2, 1/3], 否, coordinate_partials_become_a_gradient_vector_only_after_the_metric_identification)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod50-graddeep-boundary","seed":202611196090,"parameters":{"mode":"non_euclidean_metric","model":{"coordinate_gradient":[15,1],"metric":[[2,0],[0,3]]},"domain":"Exact_gradient_vector_density","operation":"graddeep-boundary","semantic_template":"graddeep-boundary|{\"mode\":\"non_euclidean_metric\",\"model\":{\"coordinate_gradient\":[[15,1],[1,1]],\"metric\":[[[2,1],[0,1]],[[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-r06-0091","batch_id":"batch-20261015-advance-20261119-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-boundary；明示資料={\"mode\":\"finite_difference_confusion\",\"model\":{\"difference_quotient\":16,\"step\":[1,17]}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(finite_difference_confusion, 16, 1/17, 否, one_finite_difference_quotient_is_not_a_derivative_or_gradient_certificate)","explanation_zh":"Wave6第十三層核對=(finite_difference_confusion, 16, 1/17, 否, one_finite_difference_quotient_is_not_a_derivative_or_gradient_certificate)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod50-graddeep-boundary","seed":202611196091,"parameters":{"mode":"finite_difference_confusion","model":{"difference_quotient":16,"step":[1,17]},"domain":"Exact_gradient_vector_density","operation":"graddeep-boundary","semantic_template":"graddeep-boundary|{\"mode\":\"finite_difference_confusion\",\"model\":{\"difference_quotient\":[16,1],\"step\":[1,17]}}"}},"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-r06-0092","batch_id":"batch-20261015-advance-20261119-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-boundary；明示資料={\"mode\":\"finite_difference_confusion\",\"model\":{\"difference_quotient\":17,\"step\":[1,18]}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(finite_difference_confusion, 17, 1/18, 否, one_finite_difference_quotient_is_not_a_derivative_or_gradient_certificate)","explanation_zh":"Wave6第十三層核對=(finite_difference_confusion, 17, 1/18, 否, one_finite_difference_quotient_is_not_a_derivative_or_gradient_certificate)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod50-graddeep-boundary","seed":202611196092,"parameters":{"mode":"finite_difference_confusion","model":{"difference_quotient":17,"step":[1,18]},"domain":"Exact_gradient_vector_density","operation":"graddeep-boundary","semantic_template":"graddeep-boundary|{\"mode\":\"finite_difference_confusion\",\"model\":{\"difference_quotient\":[17,1],\"step\":[1,18]}}"}},"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-r06-0093","batch_id":"batch-20261015-advance-20261119-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-boundary；明示資料={\"mode\":\"positive_gradient_direction\",\"model\":{\"gradient\":[18,-1]}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(positive_gradient_direction, [18, -1], [18, -1], 325, 是, 否, positive_gradient_is_first_order_ascent_not_descent)","explanation_zh":"Wave6第十三層核對=(positive_gradient_direction, [18, -1], [18, -1], 325, 是, 否, positive_gradient_is_first_order_ascent_not_descent)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod50-graddeep-boundary","seed":202611196093,"parameters":{"mode":"positive_gradient_direction","model":{"gradient":[18,-1]},"domain":"Exact_gradient_vector_density","operation":"graddeep-boundary","semantic_template":"graddeep-boundary|{\"mode\":\"positive_gradient_direction\",\"model\":{\"gradient\":[[18,1],[-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-r06-0094","batch_id":"batch-20261015-advance-20261119-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-boundary；明示資料={\"mode\":\"non_euclidean_metric\",\"model\":{\"coordinate_gradient\":[19,1],\"metric\":[[2,0],[0,3]]}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(non_euclidean_metric, [19, 1], [[2, 0], [0, 3]], [19/2, 1/3], 否, coordinate_partials_become_a_gradient_vector_only_after_the_metric_identification)","explanation_zh":"Wave6第十三層核對=(non_euclidean_metric, [19, 1], [[2, 0], [0, 3]], [19/2, 1/3], 否, coordinate_partials_become_a_gradient_vector_only_after_the_metric_identification)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod50-graddeep-boundary","seed":202611196094,"parameters":{"mode":"non_euclidean_metric","model":{"coordinate_gradient":[19,1],"metric":[[2,0],[0,3]]},"domain":"Exact_gradient_vector_density","operation":"graddeep-boundary","semantic_template":"graddeep-boundary|{\"mode\":\"non_euclidean_metric\",\"model\":{\"coordinate_gradient\":[[19,1],[1,1]],\"metric\":[[[2,1],[0,1]],[[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-r06-0095","batch_id":"batch-20261015-advance-20261119-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-boundary；明示資料={\"mode\":\"finite_difference_confusion\",\"model\":{\"difference_quotient\":20,\"step\":[1,21]}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(finite_difference_confusion, 20, 1/21, 否, one_finite_difference_quotient_is_not_a_derivative_or_gradient_certificate)","explanation_zh":"Wave6第十三層核對=(finite_difference_confusion, 20, 1/21, 否, one_finite_difference_quotient_is_not_a_derivative_or_gradient_certificate)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod50-graddeep-boundary","seed":202611196095,"parameters":{"mode":"finite_difference_confusion","model":{"difference_quotient":20,"step":[1,21]},"domain":"Exact_gradient_vector_density","operation":"graddeep-boundary","semantic_template":"graddeep-boundary|{\"mode\":\"finite_difference_confusion\",\"model\":{\"difference_quotient\":[20,1],\"step\":[1,21]}}"}},"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-r06-0096","batch_id":"batch-20261015-advance-20261119-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-boundary；明示資料={\"mode\":\"finite_difference_confusion\",\"model\":{\"difference_quotient\":21,\"step\":[1,22]}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(finite_difference_confusion, 21, 1/22, 否, one_finite_difference_quotient_is_not_a_derivative_or_gradient_certificate)","explanation_zh":"Wave6第十三層核對=(finite_difference_confusion, 21, 1/22, 否, one_finite_difference_quotient_is_not_a_derivative_or_gradient_certificate)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod50-graddeep-boundary","seed":202611196096,"parameters":{"mode":"finite_difference_confusion","model":{"difference_quotient":21,"step":[1,22]},"domain":"Exact_gradient_vector_density","operation":"graddeep-boundary","semantic_template":"graddeep-boundary|{\"mode\":\"finite_difference_confusion\",\"model\":{\"difference_quotient\":[21,1],\"step\":[1,22]}}"}},"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-r06-0097","batch_id":"batch-20261015-advance-20261119-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-boundary；明示資料={\"mode\":\"positive_gradient_direction\",\"model\":{\"gradient\":[22,-1]}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(positive_gradient_direction, [22, -1], [22, -1], 485, 是, 否, positive_gradient_is_first_order_ascent_not_descent)","explanation_zh":"Wave6第十三層核對=(positive_gradient_direction, [22, -1], [22, -1], 485, 是, 否, positive_gradient_is_first_order_ascent_not_descent)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod50-graddeep-boundary","seed":202611196097,"parameters":{"mode":"positive_gradient_direction","model":{"gradient":[22,-1]},"domain":"Exact_gradient_vector_density","operation":"graddeep-boundary","semantic_template":"graddeep-boundary|{\"mode\":\"positive_gradient_direction\",\"model\":{\"gradient\":[[22,1],[-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-r06-0098","batch_id":"batch-20261015-advance-20261119-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-boundary；明示資料={\"mode\":\"non_euclidean_metric\",\"model\":{\"coordinate_gradient\":[23,1],\"metric\":[[2,0],[0,3]]}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(non_euclidean_metric, [23, 1], [[2, 0], [0, 3]], [23/2, 1/3], 否, coordinate_partials_become_a_gradient_vector_only_after_the_metric_identification)","explanation_zh":"Wave6第十三層核對=(non_euclidean_metric, [23, 1], [[2, 0], [0, 3]], [23/2, 1/3], 否, coordinate_partials_become_a_gradient_vector_only_after_the_metric_identification)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod50-graddeep-boundary","seed":202611196098,"parameters":{"mode":"non_euclidean_metric","model":{"coordinate_gradient":[23,1],"metric":[[2,0],[0,3]]},"domain":"Exact_gradient_vector_density","operation":"graddeep-boundary","semantic_template":"graddeep-boundary|{\"mode\":\"non_euclidean_metric\",\"model\":{\"coordinate_gradient\":[[23,1],[1,1]],\"metric\":[[[2,1],[0,1]],[[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-r06-0099","batch_id":"batch-20261015-advance-20261119-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-boundary；明示資料={\"mode\":\"finite_difference_confusion\",\"model\":{\"difference_quotient\":24,\"step\":[1,25]}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(finite_difference_confusion, 24, 1/25, 否, one_finite_difference_quotient_is_not_a_derivative_or_gradient_certificate)","explanation_zh":"Wave6第十三層核對=(finite_difference_confusion, 24, 1/25, 否, one_finite_difference_quotient_is_not_a_derivative_or_gradient_certificate)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod50-graddeep-boundary","seed":202611196099,"parameters":{"mode":"finite_difference_confusion","model":{"difference_quotient":24,"step":[1,25]},"domain":"Exact_gradient_vector_density","operation":"graddeep-boundary","semantic_template":"graddeep-boundary|{\"mode\":\"finite_difference_confusion\",\"model\":{\"difference_quotient\":[24,1],\"step\":[1,25]}}"}},"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-r06-0100","batch_id":"batch-20261015-advance-20261119-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-boundary；明示資料={\"mode\":\"finite_difference_confusion\",\"model\":{\"difference_quotient\":25,\"step\":[1,26]}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(finite_difference_confusion, 25, 1/26, 否, one_finite_difference_quotient_is_not_a_derivative_or_gradient_certificate)","explanation_zh":"Wave6第十三層核對=(finite_difference_confusion, 25, 1/26, 否, one_finite_difference_quotient_is_not_a_derivative_or_gradient_certificate)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod50-graddeep-boundary","seed":202611196100,"parameters":{"mode":"finite_difference_confusion","model":{"difference_quotient":25,"step":[1,26]},"domain":"Exact_gradient_vector_density","operation":"graddeep-boundary","semantic_template":"graddeep-boundary|{\"mode\":\"finite_difference_confusion\",\"model\":{\"difference_quotient\":[25,1],\"step\":[1,26]}}"}},"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"}}