{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r06-gradient-vector-0001","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-symbolic-polynomial；資料={\"dimension\":2,\"terms\":[[1,[2,1]],[2,[0,1]],[1,[0,0]]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [[1, [0, 0]], [2, [0, 1]], [1, [2, 1]]], [[[2, [1, 1]]], [[2, [0, 0]], [1, [2, 0]]]], standard_Euclidean_coordinate_order_preserved)","explanation_zh":"Wave6第七層核對=(2, [[1, [0, 0]], [2, [0, 1]], [1, [2, 1]]], [[[2, [1, 1]]], [[2, [0, 0]], [1, [2, 0]]]], standard_Euclidean_coordinate_order_preserved)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod44-gradient-symbolic-polynomial","seed":202611136001,"parameters":{"dimension":2,"terms":[[1,[2,1]],[2,[0,1]],[1,[0,0]]],"domain":"Exact_polynomial_gradient_vector","operation":"gradient-symbolic-polynomial","semantic_template":"qf-mod44-gradient-symbolic-polynomial/{\"dimension\":2,\"terms\":[[[1,1],[0,0]],[[1,1],[2,1]],[[2,1],[0,1]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r06-gradient-vector-0002","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-symbolic-polynomial；資料={\"dimension\":3,\"terms\":[[2,[2,1,0]],[3,[0,1,1]],[1,[0,0,0]]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, [[1, [0, 0, 0]], [3, [0, 1, 1]], [2, [2, 1, 0]]], [[[4, [1, 1, 0]]], [[3, [0, 0, 1]], [2, [2, 0, 0]]], [[3, [0, 1, 0]]]], standard_Euclidean_coordinate_order_preserved)","explanation_zh":"Wave6第七層核對=(3, [[1, [0, 0, 0]], [3, [0, 1, 1]], [2, [2, 1, 0]]], [[[4, [1, 1, 0]]], [[3, [0, 0, 1]], [2, [2, 0, 0]]], [[3, [0, 1, 0]]]], standard_Euclidean_coordinate_order_preserved)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod44-gradient-symbolic-polynomial","seed":202611136002,"parameters":{"dimension":3,"terms":[[2,[2,1,0]],[3,[0,1,1]],[1,[0,0,0]]],"domain":"Exact_polynomial_gradient_vector","operation":"gradient-symbolic-polynomial","semantic_template":"qf-mod44-gradient-symbolic-polynomial/{\"dimension\":3,\"terms\":[[[1,1],[0,0,0]],[[2,1],[2,1,0]],[[3,1],[0,1,1]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r06-gradient-vector-0003","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-symbolic-polynomial；資料={\"dimension\":2,\"terms\":[[3,[2,1]],[4,[0,1]],[1,[0,0]]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [[1, [0, 0]], [4, [0, 1]], [3, [2, 1]]], [[[6, [1, 1]]], [[4, [0, 0]], [3, [2, 0]]]], standard_Euclidean_coordinate_order_preserved)","explanation_zh":"Wave6第七層核對=(2, [[1, [0, 0]], [4, [0, 1]], [3, [2, 1]]], [[[6, [1, 1]]], [[4, [0, 0]], [3, [2, 0]]]], standard_Euclidean_coordinate_order_preserved)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod44-gradient-symbolic-polynomial","seed":202611136003,"parameters":{"dimension":2,"terms":[[3,[2,1]],[4,[0,1]],[1,[0,0]]],"domain":"Exact_polynomial_gradient_vector","operation":"gradient-symbolic-polynomial","semantic_template":"qf-mod44-gradient-symbolic-polynomial/{\"dimension\":2,\"terms\":[[[1,1],[0,0]],[[3,1],[2,1]],[[4,1],[0,1]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r06-gradient-vector-0004","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-symbolic-polynomial；資料={\"dimension\":3,\"terms\":[[4,[2,1,0]],[5,[0,1,1]],[1,[0,0,0]]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, [[1, [0, 0, 0]], [5, [0, 1, 1]], [4, [2, 1, 0]]], [[[8, [1, 1, 0]]], [[5, [0, 0, 1]], [4, [2, 0, 0]]], [[5, [0, 1, 0]]]], standard_Euclidean_coordinate_order_preserved)","explanation_zh":"Wave6第七層核對=(3, [[1, [0, 0, 0]], [5, [0, 1, 1]], [4, [2, 1, 0]]], [[[8, [1, 1, 0]]], [[5, [0, 0, 1]], [4, [2, 0, 0]]], [[5, [0, 1, 0]]]], standard_Euclidean_coordinate_order_preserved)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod44-gradient-symbolic-polynomial","seed":202611136004,"parameters":{"dimension":3,"terms":[[4,[2,1,0]],[5,[0,1,1]],[1,[0,0,0]]],"domain":"Exact_polynomial_gradient_vector","operation":"gradient-symbolic-polynomial","semantic_template":"qf-mod44-gradient-symbolic-polynomial/{\"dimension\":3,\"terms\":[[[1,1],[0,0,0]],[[4,1],[2,1,0]],[[5,1],[0,1,1]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r06-gradient-vector-0005","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-symbolic-polynomial；資料={\"dimension\":2,\"terms\":[[5,[2,1]],[6,[0,1]],[1,[0,0]]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [[1, [0, 0]], [6, [0, 1]], [5, [2, 1]]], [[[10, [1, 1]]], [[6, [0, 0]], [5, [2, 0]]]], standard_Euclidean_coordinate_order_preserved)","explanation_zh":"Wave6第七層核對=(2, [[1, [0, 0]], [6, [0, 1]], [5, [2, 1]]], [[[10, [1, 1]]], [[6, [0, 0]], [5, [2, 0]]]], standard_Euclidean_coordinate_order_preserved)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod44-gradient-symbolic-polynomial","seed":202611136005,"parameters":{"dimension":2,"terms":[[5,[2,1]],[6,[0,1]],[1,[0,0]]],"domain":"Exact_polynomial_gradient_vector","operation":"gradient-symbolic-polynomial","semantic_template":"qf-mod44-gradient-symbolic-polynomial/{\"dimension\":2,\"terms\":[[[1,1],[0,0]],[[5,1],[2,1]],[[6,1],[0,1]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r06-gradient-vector-0006","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-symbolic-polynomial；資料={\"dimension\":3,\"terms\":[[6,[2,1,0]],[7,[0,1,1]],[1,[0,0,0]]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, [[1, [0, 0, 0]], [7, [0, 1, 1]], [6, [2, 1, 0]]], [[[12, [1, 1, 0]]], [[7, [0, 0, 1]], [6, [2, 0, 0]]], [[7, [0, 1, 0]]]], standard_Euclidean_coordinate_order_preserved)","explanation_zh":"Wave6第七層核對=(3, [[1, [0, 0, 0]], [7, [0, 1, 1]], [6, [2, 1, 0]]], [[[12, [1, 1, 0]]], [[7, [0, 0, 1]], [6, [2, 0, 0]]], [[7, [0, 1, 0]]]], standard_Euclidean_coordinate_order_preserved)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod44-gradient-symbolic-polynomial","seed":202611136006,"parameters":{"dimension":3,"terms":[[6,[2,1,0]],[7,[0,1,1]],[1,[0,0,0]]],"domain":"Exact_polynomial_gradient_vector","operation":"gradient-symbolic-polynomial","semantic_template":"qf-mod44-gradient-symbolic-polynomial/{\"dimension\":3,\"terms\":[[[1,1],[0,0,0]],[[6,1],[2,1,0]],[[7,1],[0,1,1]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r06-gradient-vector-0007","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-symbolic-polynomial；資料={\"dimension\":2,\"terms\":[[7,[2,1]],[8,[0,1]],[1,[0,0]]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [[1, [0, 0]], [8, [0, 1]], [7, [2, 1]]], [[[14, [1, 1]]], [[8, [0, 0]], [7, [2, 0]]]], standard_Euclidean_coordinate_order_preserved)","explanation_zh":"Wave6第七層核對=(2, [[1, [0, 0]], [8, [0, 1]], [7, [2, 1]]], [[[14, [1, 1]]], [[8, [0, 0]], [7, [2, 0]]]], standard_Euclidean_coordinate_order_preserved)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod44-gradient-symbolic-polynomial","seed":202611136007,"parameters":{"dimension":2,"terms":[[7,[2,1]],[8,[0,1]],[1,[0,0]]],"domain":"Exact_polynomial_gradient_vector","operation":"gradient-symbolic-polynomial","semantic_template":"qf-mod44-gradient-symbolic-polynomial/{\"dimension\":2,\"terms\":[[[1,1],[0,0]],[[7,1],[2,1]],[[8,1],[0,1]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r06-gradient-vector-0008","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-symbolic-polynomial；資料={\"dimension\":3,\"terms\":[[8,[2,1,0]],[9,[0,1,1]],[1,[0,0,0]]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, [[1, [0, 0, 0]], [9, [0, 1, 1]], [8, [2, 1, 0]]], [[[16, [1, 1, 0]]], [[9, [0, 0, 1]], [8, [2, 0, 0]]], [[9, [0, 1, 0]]]], standard_Euclidean_coordinate_order_preserved)","explanation_zh":"Wave6第七層核對=(3, [[1, [0, 0, 0]], [9, [0, 1, 1]], [8, [2, 1, 0]]], [[[16, [1, 1, 0]]], [[9, [0, 0, 1]], [8, [2, 0, 0]]], [[9, [0, 1, 0]]]], standard_Euclidean_coordinate_order_preserved)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod44-gradient-symbolic-polynomial","seed":202611136008,"parameters":{"dimension":3,"terms":[[8,[2,1,0]],[9,[0,1,1]],[1,[0,0,0]]],"domain":"Exact_polynomial_gradient_vector","operation":"gradient-symbolic-polynomial","semantic_template":"qf-mod44-gradient-symbolic-polynomial/{\"dimension\":3,\"terms\":[[[1,1],[0,0,0]],[[8,1],[2,1,0]],[[9,1],[0,1,1]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r06-gradient-vector-0009","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-symbolic-polynomial；資料={\"dimension\":2,\"terms\":[[9,[2,1]],[10,[0,1]],[1,[0,0]]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [[1, [0, 0]], [10, [0, 1]], [9, [2, 1]]], [[[18, [1, 1]]], [[10, [0, 0]], [9, [2, 0]]]], standard_Euclidean_coordinate_order_preserved)","explanation_zh":"Wave6第七層核對=(2, [[1, [0, 0]], [10, [0, 1]], [9, [2, 1]]], [[[18, [1, 1]]], [[10, [0, 0]], [9, [2, 0]]]], standard_Euclidean_coordinate_order_preserved)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod44-gradient-symbolic-polynomial","seed":202611136009,"parameters":{"dimension":2,"terms":[[9,[2,1]],[10,[0,1]],[1,[0,0]]],"domain":"Exact_polynomial_gradient_vector","operation":"gradient-symbolic-polynomial","semantic_template":"qf-mod44-gradient-symbolic-polynomial/{\"dimension\":2,\"terms\":[[[1,1],[0,0]],[[10,1],[0,1]],[[9,1],[2,1]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r06-gradient-vector-0010","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-symbolic-polynomial；資料={\"dimension\":3,\"terms\":[[10,[2,1,0]],[11,[0,1,1]],[1,[0,0,0]]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, [[1, [0, 0, 0]], [11, [0, 1, 1]], [10, [2, 1, 0]]], [[[20, [1, 1, 0]]], [[11, [0, 0, 1]], [10, [2, 0, 0]]], [[11, [0, 1, 0]]]], standard_Euclidean_coordinate_order_preserved)","explanation_zh":"Wave6第七層核對=(3, [[1, [0, 0, 0]], [11, [0, 1, 1]], [10, [2, 1, 0]]], [[[20, [1, 1, 0]]], [[11, [0, 0, 1]], [10, [2, 0, 0]]], [[11, [0, 1, 0]]]], standard_Euclidean_coordinate_order_preserved)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod44-gradient-symbolic-polynomial","seed":202611136010,"parameters":{"dimension":3,"terms":[[10,[2,1,0]],[11,[0,1,1]],[1,[0,0,0]]],"domain":"Exact_polynomial_gradient_vector","operation":"gradient-symbolic-polynomial","semantic_template":"qf-mod44-gradient-symbolic-polynomial/{\"dimension\":3,\"terms\":[[[1,1],[0,0,0]],[[10,1],[2,1,0]],[[11,1],[0,1,1]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r06-gradient-vector-0011","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-symbolic-polynomial；資料={\"dimension\":2,\"terms\":[[11,[2,1]],[12,[0,1]],[1,[0,0]]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [[1, [0, 0]], [12, [0, 1]], [11, [2, 1]]], [[[22, [1, 1]]], [[12, [0, 0]], [11, [2, 0]]]], standard_Euclidean_coordinate_order_preserved)","explanation_zh":"Wave6第七層核對=(2, [[1, [0, 0]], [12, [0, 1]], [11, [2, 1]]], [[[22, [1, 1]]], [[12, [0, 0]], [11, [2, 0]]]], standard_Euclidean_coordinate_order_preserved)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod44-gradient-symbolic-polynomial","seed":202611136011,"parameters":{"dimension":2,"terms":[[11,[2,1]],[12,[0,1]],[1,[0,0]]],"domain":"Exact_polynomial_gradient_vector","operation":"gradient-symbolic-polynomial","semantic_template":"qf-mod44-gradient-symbolic-polynomial/{\"dimension\":2,\"terms\":[[[1,1],[0,0]],[[11,1],[2,1]],[[12,1],[0,1]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r06-gradient-vector-0012","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-symbolic-polynomial；資料={\"dimension\":3,\"terms\":[[12,[2,1,0]],[13,[0,1,1]],[1,[0,0,0]]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, [[1, [0, 0, 0]], [13, [0, 1, 1]], [12, [2, 1, 0]]], [[[24, [1, 1, 0]]], [[13, [0, 0, 1]], [12, [2, 0, 0]]], [[13, [0, 1, 0]]]], standard_Euclidean_coordinate_order_preserved)","explanation_zh":"Wave6第七層核對=(3, [[1, [0, 0, 0]], [13, [0, 1, 1]], [12, [2, 1, 0]]], [[[24, [1, 1, 0]]], [[13, [0, 0, 1]], [12, [2, 0, 0]]], [[13, [0, 1, 0]]]], standard_Euclidean_coordinate_order_preserved)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod44-gradient-symbolic-polynomial","seed":202611136012,"parameters":{"dimension":3,"terms":[[12,[2,1,0]],[13,[0,1,1]],[1,[0,0,0]]],"domain":"Exact_polynomial_gradient_vector","operation":"gradient-symbolic-polynomial","semantic_template":"qf-mod44-gradient-symbolic-polynomial/{\"dimension\":3,\"terms\":[[[1,1],[0,0,0]],[[12,1],[2,1,0]],[[13,1],[0,1,1]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r06-gradient-vector-0013","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-symbolic-polynomial；資料={\"dimension\":2,\"terms\":[[13,[2,1]],[14,[0,1]],[1,[0,0]]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [[1, [0, 0]], [14, [0, 1]], [13, [2, 1]]], [[[26, [1, 1]]], [[14, [0, 0]], [13, [2, 0]]]], standard_Euclidean_coordinate_order_preserved)","explanation_zh":"Wave6第七層核對=(2, [[1, [0, 0]], [14, [0, 1]], [13, [2, 1]]], [[[26, [1, 1]]], [[14, [0, 0]], [13, [2, 0]]]], standard_Euclidean_coordinate_order_preserved)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod44-gradient-symbolic-polynomial","seed":202611136013,"parameters":{"dimension":2,"terms":[[13,[2,1]],[14,[0,1]],[1,[0,0]]],"domain":"Exact_polynomial_gradient_vector","operation":"gradient-symbolic-polynomial","semantic_template":"qf-mod44-gradient-symbolic-polynomial/{\"dimension\":2,\"terms\":[[[1,1],[0,0]],[[13,1],[2,1]],[[14,1],[0,1]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r06-gradient-vector-0014","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-symbolic-polynomial；資料={\"dimension\":3,\"terms\":[[14,[2,1,0]],[15,[0,1,1]],[1,[0,0,0]]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, [[1, [0, 0, 0]], [15, [0, 1, 1]], [14, [2, 1, 0]]], [[[28, [1, 1, 0]]], [[15, [0, 0, 1]], [14, [2, 0, 0]]], [[15, [0, 1, 0]]]], standard_Euclidean_coordinate_order_preserved)","explanation_zh":"Wave6第七層核對=(3, [[1, [0, 0, 0]], [15, [0, 1, 1]], [14, [2, 1, 0]]], [[[28, [1, 1, 0]]], [[15, [0, 0, 1]], [14, [2, 0, 0]]], [[15, [0, 1, 0]]]], standard_Euclidean_coordinate_order_preserved)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod44-gradient-symbolic-polynomial","seed":202611136014,"parameters":{"dimension":3,"terms":[[14,[2,1,0]],[15,[0,1,1]],[1,[0,0,0]]],"domain":"Exact_polynomial_gradient_vector","operation":"gradient-symbolic-polynomial","semantic_template":"qf-mod44-gradient-symbolic-polynomial/{\"dimension\":3,\"terms\":[[[1,1],[0,0,0]],[[14,1],[2,1,0]],[[15,1],[0,1,1]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r06-gradient-vector-0015","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-symbolic-polynomial；資料={\"dimension\":2,\"terms\":[[15,[2,1]],[16,[0,1]],[1,[0,0]]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [[1, [0, 0]], [16, [0, 1]], [15, [2, 1]]], [[[30, [1, 1]]], [[16, [0, 0]], [15, [2, 0]]]], standard_Euclidean_coordinate_order_preserved)","explanation_zh":"Wave6第七層核對=(2, [[1, [0, 0]], [16, [0, 1]], [15, [2, 1]]], [[[30, [1, 1]]], [[16, [0, 0]], [15, [2, 0]]]], standard_Euclidean_coordinate_order_preserved)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod44-gradient-symbolic-polynomial","seed":202611136015,"parameters":{"dimension":2,"terms":[[15,[2,1]],[16,[0,1]],[1,[0,0]]],"domain":"Exact_polynomial_gradient_vector","operation":"gradient-symbolic-polynomial","semantic_template":"qf-mod44-gradient-symbolic-polynomial/{\"dimension\":2,\"terms\":[[[1,1],[0,0]],[[15,1],[2,1]],[[16,1],[0,1]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r06-gradient-vector-0016","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-symbolic-polynomial；資料={\"dimension\":3,\"terms\":[[16,[2,1,0]],[17,[0,1,1]],[1,[0,0,0]]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, [[1, [0, 0, 0]], [17, [0, 1, 1]], [16, [2, 1, 0]]], [[[32, [1, 1, 0]]], [[17, [0, 0, 1]], [16, [2, 0, 0]]], [[17, [0, 1, 0]]]], standard_Euclidean_coordinate_order_preserved)","explanation_zh":"Wave6第七層核對=(3, [[1, [0, 0, 0]], [17, [0, 1, 1]], [16, [2, 1, 0]]], [[[32, [1, 1, 0]]], [[17, [0, 0, 1]], [16, [2, 0, 0]]], [[17, [0, 1, 0]]]], standard_Euclidean_coordinate_order_preserved)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod44-gradient-symbolic-polynomial","seed":202611136016,"parameters":{"dimension":3,"terms":[[16,[2,1,0]],[17,[0,1,1]],[1,[0,0,0]]],"domain":"Exact_polynomial_gradient_vector","operation":"gradient-symbolic-polynomial","semantic_template":"qf-mod44-gradient-symbolic-polynomial/{\"dimension\":3,\"terms\":[[[1,1],[0,0,0]],[[16,1],[2,1,0]],[[17,1],[0,1,1]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r06-gradient-vector-0017","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-symbolic-polynomial；資料={\"dimension\":2,\"terms\":[[17,[2,1]],[18,[0,1]],[1,[0,0]]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [[1, [0, 0]], [18, [0, 1]], [17, [2, 1]]], [[[34, [1, 1]]], [[18, [0, 0]], [17, [2, 0]]]], standard_Euclidean_coordinate_order_preserved)","explanation_zh":"Wave6第七層核對=(2, [[1, [0, 0]], [18, [0, 1]], [17, [2, 1]]], [[[34, [1, 1]]], [[18, [0, 0]], [17, [2, 0]]]], standard_Euclidean_coordinate_order_preserved)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod44-gradient-symbolic-polynomial","seed":202611136017,"parameters":{"dimension":2,"terms":[[17,[2,1]],[18,[0,1]],[1,[0,0]]],"domain":"Exact_polynomial_gradient_vector","operation":"gradient-symbolic-polynomial","semantic_template":"qf-mod44-gradient-symbolic-polynomial/{\"dimension\":2,\"terms\":[[[1,1],[0,0]],[[17,1],[2,1]],[[18,1],[0,1]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r06-gradient-vector-0018","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-symbolic-polynomial；資料={\"dimension\":3,\"terms\":[[18,[2,1,0]],[19,[0,1,1]],[1,[0,0,0]]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, [[1, [0, 0, 0]], [19, [0, 1, 1]], [18, [2, 1, 0]]], [[[36, [1, 1, 0]]], [[19, [0, 0, 1]], [18, [2, 0, 0]]], [[19, [0, 1, 0]]]], standard_Euclidean_coordinate_order_preserved)","explanation_zh":"Wave6第七層核對=(3, [[1, [0, 0, 0]], [19, [0, 1, 1]], [18, [2, 1, 0]]], [[[36, [1, 1, 0]]], [[19, [0, 0, 1]], [18, [2, 0, 0]]], [[19, [0, 1, 0]]]], standard_Euclidean_coordinate_order_preserved)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod44-gradient-symbolic-polynomial","seed":202611136018,"parameters":{"dimension":3,"terms":[[18,[2,1,0]],[19,[0,1,1]],[1,[0,0,0]]],"domain":"Exact_polynomial_gradient_vector","operation":"gradient-symbolic-polynomial","semantic_template":"qf-mod44-gradient-symbolic-polynomial/{\"dimension\":3,\"terms\":[[[1,1],[0,0,0]],[[18,1],[2,1,0]],[[19,1],[0,1,1]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r06-gradient-vector-0019","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-symbolic-polynomial；資料={\"dimension\":2,\"terms\":[[19,[2,1]],[20,[0,1]],[1,[0,0]]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [[1, [0, 0]], [20, [0, 1]], [19, [2, 1]]], [[[38, [1, 1]]], [[20, [0, 0]], [19, [2, 0]]]], standard_Euclidean_coordinate_order_preserved)","explanation_zh":"Wave6第七層核對=(2, [[1, [0, 0]], [20, [0, 1]], [19, [2, 1]]], [[[38, [1, 1]]], [[20, [0, 0]], [19, [2, 0]]]], standard_Euclidean_coordinate_order_preserved)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod44-gradient-symbolic-polynomial","seed":202611136019,"parameters":{"dimension":2,"terms":[[19,[2,1]],[20,[0,1]],[1,[0,0]]],"domain":"Exact_polynomial_gradient_vector","operation":"gradient-symbolic-polynomial","semantic_template":"qf-mod44-gradient-symbolic-polynomial/{\"dimension\":2,\"terms\":[[[1,1],[0,0]],[[19,1],[2,1]],[[20,1],[0,1]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r06-gradient-vector-0020","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-symbolic-polynomial；資料={\"dimension\":3,\"terms\":[[20,[2,1,0]],[21,[0,1,1]],[1,[0,0,0]]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, [[1, [0, 0, 0]], [21, [0, 1, 1]], [20, [2, 1, 0]]], [[[40, [1, 1, 0]]], [[21, [0, 0, 1]], [20, [2, 0, 0]]], [[21, [0, 1, 0]]]], standard_Euclidean_coordinate_order_preserved)","explanation_zh":"Wave6第七層核對=(3, [[1, [0, 0, 0]], [21, [0, 1, 1]], [20, [2, 1, 0]]], [[[40, [1, 1, 0]]], [[21, [0, 0, 1]], [20, [2, 0, 0]]], [[21, [0, 1, 0]]]], standard_Euclidean_coordinate_order_preserved)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod44-gradient-symbolic-polynomial","seed":202611136020,"parameters":{"dimension":3,"terms":[[20,[2,1,0]],[21,[0,1,1]],[1,[0,0,0]]],"domain":"Exact_polynomial_gradient_vector","operation":"gradient-symbolic-polynomial","semantic_template":"qf-mod44-gradient-symbolic-polynomial/{\"dimension\":3,\"terms\":[[[1,1],[0,0,0]],[[20,1],[2,1,0]],[[21,1],[0,1,1]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r06-gradient-vector-0021","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-symbolic-polynomial；資料={\"dimension\":2,\"terms\":[[21,[2,1]],[22,[0,1]],[1,[0,0]]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [[1, [0, 0]], [22, [0, 1]], [21, [2, 1]]], [[[42, [1, 1]]], [[22, [0, 0]], [21, [2, 0]]]], standard_Euclidean_coordinate_order_preserved)","explanation_zh":"Wave6第七層核對=(2, [[1, [0, 0]], [22, [0, 1]], [21, [2, 1]]], [[[42, [1, 1]]], [[22, [0, 0]], [21, [2, 0]]]], standard_Euclidean_coordinate_order_preserved)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod44-gradient-symbolic-polynomial","seed":202611136021,"parameters":{"dimension":2,"terms":[[21,[2,1]],[22,[0,1]],[1,[0,0]]],"domain":"Exact_polynomial_gradient_vector","operation":"gradient-symbolic-polynomial","semantic_template":"qf-mod44-gradient-symbolic-polynomial/{\"dimension\":2,\"terms\":[[[1,1],[0,0]],[[21,1],[2,1]],[[22,1],[0,1]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r06-gradient-vector-0022","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-symbolic-polynomial；資料={\"dimension\":3,\"terms\":[[22,[2,1,0]],[23,[0,1,1]],[1,[0,0,0]]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, [[1, [0, 0, 0]], [23, [0, 1, 1]], [22, [2, 1, 0]]], [[[44, [1, 1, 0]]], [[23, [0, 0, 1]], [22, [2, 0, 0]]], [[23, [0, 1, 0]]]], standard_Euclidean_coordinate_order_preserved)","explanation_zh":"Wave6第七層核對=(3, [[1, [0, 0, 0]], [23, [0, 1, 1]], [22, [2, 1, 0]]], [[[44, [1, 1, 0]]], [[23, [0, 0, 1]], [22, [2, 0, 0]]], [[23, [0, 1, 0]]]], standard_Euclidean_coordinate_order_preserved)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod44-gradient-symbolic-polynomial","seed":202611136022,"parameters":{"dimension":3,"terms":[[22,[2,1,0]],[23,[0,1,1]],[1,[0,0,0]]],"domain":"Exact_polynomial_gradient_vector","operation":"gradient-symbolic-polynomial","semantic_template":"qf-mod44-gradient-symbolic-polynomial/{\"dimension\":3,\"terms\":[[[1,1],[0,0,0]],[[22,1],[2,1,0]],[[23,1],[0,1,1]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r06-gradient-vector-0023","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-symbolic-polynomial；資料={\"dimension\":2,\"terms\":[[23,[2,1]],[24,[0,1]],[1,[0,0]]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [[1, [0, 0]], [24, [0, 1]], [23, [2, 1]]], [[[46, [1, 1]]], [[24, [0, 0]], [23, [2, 0]]]], standard_Euclidean_coordinate_order_preserved)","explanation_zh":"Wave6第七層核對=(2, [[1, [0, 0]], [24, [0, 1]], [23, [2, 1]]], [[[46, [1, 1]]], [[24, [0, 0]], [23, [2, 0]]]], standard_Euclidean_coordinate_order_preserved)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod44-gradient-symbolic-polynomial","seed":202611136023,"parameters":{"dimension":2,"terms":[[23,[2,1]],[24,[0,1]],[1,[0,0]]],"domain":"Exact_polynomial_gradient_vector","operation":"gradient-symbolic-polynomial","semantic_template":"qf-mod44-gradient-symbolic-polynomial/{\"dimension\":2,\"terms\":[[[1,1],[0,0]],[[23,1],[2,1]],[[24,1],[0,1]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r06-gradient-vector-0024","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-symbolic-polynomial；資料={\"dimension\":3,\"terms\":[[24,[2,1,0]],[25,[0,1,1]],[1,[0,0,0]]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, [[1, [0, 0, 0]], [25, [0, 1, 1]], [24, [2, 1, 0]]], [[[48, [1, 1, 0]]], [[25, [0, 0, 1]], [24, [2, 0, 0]]], [[25, [0, 1, 0]]]], standard_Euclidean_coordinate_order_preserved)","explanation_zh":"Wave6第七層核對=(3, [[1, [0, 0, 0]], [25, [0, 1, 1]], [24, [2, 1, 0]]], [[[48, [1, 1, 0]]], [[25, [0, 0, 1]], [24, [2, 0, 0]]], [[25, [0, 1, 0]]]], standard_Euclidean_coordinate_order_preserved)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod44-gradient-symbolic-polynomial","seed":202611136024,"parameters":{"dimension":3,"terms":[[24,[2,1,0]],[25,[0,1,1]],[1,[0,0,0]]],"domain":"Exact_polynomial_gradient_vector","operation":"gradient-symbolic-polynomial","semantic_template":"qf-mod44-gradient-symbolic-polynomial/{\"dimension\":3,\"terms\":[[[1,1],[0,0,0]],[[24,1],[2,1,0]],[[25,1],[0,1,1]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r06-gradient-vector-0025","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-symbolic-polynomial；資料={\"dimension\":2,\"terms\":[[25,[2,1]],[26,[0,1]],[1,[0,0]]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [[1, [0, 0]], [26, [0, 1]], [25, [2, 1]]], [[[50, [1, 1]]], [[26, [0, 0]], [25, [2, 0]]]], standard_Euclidean_coordinate_order_preserved)","explanation_zh":"Wave6第七層核對=(2, [[1, [0, 0]], [26, [0, 1]], [25, [2, 1]]], [[[50, [1, 1]]], [[26, [0, 0]], [25, [2, 0]]]], standard_Euclidean_coordinate_order_preserved)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod44-gradient-symbolic-polynomial","seed":202611136025,"parameters":{"dimension":2,"terms":[[25,[2,1]],[26,[0,1]],[1,[0,0]]],"domain":"Exact_polynomial_gradient_vector","operation":"gradient-symbolic-polynomial","semantic_template":"qf-mod44-gradient-symbolic-polynomial/{\"dimension\":2,\"terms\":[[[1,1],[0,0]],[[25,1],[2,1]],[[26,1],[0,1]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r06-gradient-vector-0026","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-rational-point；資料={\"dimension\":2,\"terms\":[[1,[2,1]],[2,[0,1]],[1,[0,0]]],\"point\":[[1,1],[2,2]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [[1, [0, 0]], [2, [0, 1]], [1, [2, 1]]], [1, 1], [[[2, [1, 1]]], [[2, [0, 0]], [1, [2, 0]]]], [2, 3], exact_rational_coordinate_evaluation)","explanation_zh":"Wave6第七層核對=(2, [[1, [0, 0]], [2, [0, 1]], [1, [2, 1]]], [1, 1], [[[2, [1, 1]]], [[2, [0, 0]], [1, [2, 0]]]], [2, 3], exact_rational_coordinate_evaluation)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod44-gradient-rational-point","seed":202611136026,"parameters":{"dimension":2,"terms":[[1,[2,1]],[2,[0,1]],[1,[0,0]]],"point":[[1,1],[2,2]],"domain":"Exact_polynomial_gradient_vector","operation":"gradient-rational-point","semantic_template":"qf-mod44-gradient-rational-point/{\"dimension\":2,\"point\":[[1,1],[1,1]],\"terms\":[[[1,1],[0,0]],[[1,1],[2,1]],[[2,1],[0,1]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r06-gradient-vector-0027","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-rational-point；資料={\"dimension\":3,\"terms\":[[2,[2,1,0]],[3,[0,1,1]],[1,[0,0,0]]],\"point\":[[2,1],[3,2],[4,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, [[1, [0, 0, 0]], [3, [0, 1, 1]], [2, [2, 1, 0]]], [2, 3/2, 4/3], [[[4, [1, 1, 0]]], [[3, [0, 0, 1]], [2, [2, 0, 0]]], [[3, [0, 1, 0]]]], [12, 12, 9/2], exact_rational_coordinate_evaluation)","explanation_zh":"Wave6第七層核對=(3, [[1, [0, 0, 0]], [3, [0, 1, 1]], [2, [2, 1, 0]]], [2, 3/2, 4/3], [[[4, [1, 1, 0]]], [[3, [0, 0, 1]], [2, [2, 0, 0]]], [[3, [0, 1, 0]]]], [12, 12, 9/2], exact_rational_coordinate_evaluation)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod44-gradient-rational-point","seed":202611136027,"parameters":{"dimension":3,"terms":[[2,[2,1,0]],[3,[0,1,1]],[1,[0,0,0]]],"point":[[2,1],[3,2],[4,3]],"domain":"Exact_polynomial_gradient_vector","operation":"gradient-rational-point","semantic_template":"qf-mod44-gradient-rational-point/{\"dimension\":3,\"point\":[[2,1],[3,2],[4,3]],\"terms\":[[[1,1],[0,0,0]],[[2,1],[2,1,0]],[[3,1],[0,1,1]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r06-gradient-vector-0028","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-rational-point；資料={\"dimension\":2,\"terms\":[[3,[2,1]],[4,[0,1]],[1,[0,0]]],\"point\":[[3,1],[4,2]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [[1, [0, 0]], [4, [0, 1]], [3, [2, 1]]], [3, 2], [[[6, [1, 1]]], [[4, [0, 0]], [3, [2, 0]]]], [36, 31], exact_rational_coordinate_evaluation)","explanation_zh":"Wave6第七層核對=(2, [[1, [0, 0]], [4, [0, 1]], [3, [2, 1]]], [3, 2], [[[6, [1, 1]]], [[4, [0, 0]], [3, [2, 0]]]], [36, 31], exact_rational_coordinate_evaluation)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod44-gradient-rational-point","seed":202611136028,"parameters":{"dimension":2,"terms":[[3,[2,1]],[4,[0,1]],[1,[0,0]]],"point":[[3,1],[4,2]],"domain":"Exact_polynomial_gradient_vector","operation":"gradient-rational-point","semantic_template":"qf-mod44-gradient-rational-point/{\"dimension\":2,\"point\":[[3,1],[2,1]],\"terms\":[[[1,1],[0,0]],[[3,1],[2,1]],[[4,1],[0,1]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r06-gradient-vector-0029","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-rational-point；資料={\"dimension\":3,\"terms\":[[4,[2,1,0]],[5,[0,1,1]],[1,[0,0,0]]],\"point\":[[4,1],[5,2],[6,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, [[1, [0, 0, 0]], [5, [0, 1, 1]], [4, [2, 1, 0]]], [4, 5/2, 2], [[[8, [1, 1, 0]]], [[5, [0, 0, 1]], [4, [2, 0, 0]]], [[5, [0, 1, 0]]]], [80, 74, 25/2], exact_rational_coordinate_evaluation)","explanation_zh":"Wave6第七層核對=(3, [[1, [0, 0, 0]], [5, [0, 1, 1]], [4, [2, 1, 0]]], [4, 5/2, 2], [[[8, [1, 1, 0]]], [[5, [0, 0, 1]], [4, [2, 0, 0]]], [[5, [0, 1, 0]]]], [80, 74, 25/2], exact_rational_coordinate_evaluation)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod44-gradient-rational-point","seed":202611136029,"parameters":{"dimension":3,"terms":[[4,[2,1,0]],[5,[0,1,1]],[1,[0,0,0]]],"point":[[4,1],[5,2],[6,3]],"domain":"Exact_polynomial_gradient_vector","operation":"gradient-rational-point","semantic_template":"qf-mod44-gradient-rational-point/{\"dimension\":3,\"point\":[[4,1],[5,2],[2,1]],\"terms\":[[[1,1],[0,0,0]],[[4,1],[2,1,0]],[[5,1],[0,1,1]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r06-gradient-vector-0030","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-rational-point；資料={\"dimension\":2,\"terms\":[[5,[2,1]],[6,[0,1]],[1,[0,0]]],\"point\":[[5,1],[6,2]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [[1, [0, 0]], [6, [0, 1]], [5, [2, 1]]], [5, 3], [[[10, [1, 1]]], [[6, [0, 0]], [5, [2, 0]]]], [150, 131], exact_rational_coordinate_evaluation)","explanation_zh":"Wave6第七層核對=(2, [[1, [0, 0]], [6, [0, 1]], [5, [2, 1]]], [5, 3], [[[10, [1, 1]]], [[6, [0, 0]], [5, [2, 0]]]], [150, 131], exact_rational_coordinate_evaluation)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod44-gradient-rational-point","seed":202611136030,"parameters":{"dimension":2,"terms":[[5,[2,1]],[6,[0,1]],[1,[0,0]]],"point":[[5,1],[6,2]],"domain":"Exact_polynomial_gradient_vector","operation":"gradient-rational-point","semantic_template":"qf-mod44-gradient-rational-point/{\"dimension\":2,\"point\":[[5,1],[3,1]],\"terms\":[[[1,1],[0,0]],[[5,1],[2,1]],[[6,1],[0,1]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r06-gradient-vector-0031","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-rational-point；資料={\"dimension\":3,\"terms\":[[6,[2,1,0]],[7,[0,1,1]],[1,[0,0,0]]],\"point\":[[6,1],[7,2],[8,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, [[1, [0, 0, 0]], [7, [0, 1, 1]], [6, [2, 1, 0]]], [6, 7/2, 8/3], [[[12, [1, 1, 0]]], [[7, [0, 0, 1]], [6, [2, 0, 0]]], [[7, [0, 1, 0]]]], [252, 704/3, 49/2], exact_rational_coordinate_evaluation)","explanation_zh":"Wave6第七層核對=(3, [[1, [0, 0, 0]], [7, [0, 1, 1]], [6, [2, 1, 0]]], [6, 7/2, 8/3], [[[12, [1, 1, 0]]], [[7, [0, 0, 1]], [6, [2, 0, 0]]], [[7, [0, 1, 0]]]], [252, 704/3, 49/2], exact_rational_coordinate_evaluation)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod44-gradient-rational-point","seed":202611136031,"parameters":{"dimension":3,"terms":[[6,[2,1,0]],[7,[0,1,1]],[1,[0,0,0]]],"point":[[6,1],[7,2],[8,3]],"domain":"Exact_polynomial_gradient_vector","operation":"gradient-rational-point","semantic_template":"qf-mod44-gradient-rational-point/{\"dimension\":3,\"point\":[[6,1],[7,2],[8,3]],\"terms\":[[[1,1],[0,0,0]],[[6,1],[2,1,0]],[[7,1],[0,1,1]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r06-gradient-vector-0032","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-rational-point；資料={\"dimension\":2,\"terms\":[[7,[2,1]],[8,[0,1]],[1,[0,0]]],\"point\":[[7,1],[8,2]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [[1, [0, 0]], [8, [0, 1]], [7, [2, 1]]], [7, 4], [[[14, [1, 1]]], [[8, [0, 0]], [7, [2, 0]]]], [392, 351], exact_rational_coordinate_evaluation)","explanation_zh":"Wave6第七層核對=(2, [[1, [0, 0]], [8, [0, 1]], [7, [2, 1]]], [7, 4], [[[14, [1, 1]]], [[8, [0, 0]], [7, [2, 0]]]], [392, 351], exact_rational_coordinate_evaluation)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod44-gradient-rational-point","seed":202611136032,"parameters":{"dimension":2,"terms":[[7,[2,1]],[8,[0,1]],[1,[0,0]]],"point":[[7,1],[8,2]],"domain":"Exact_polynomial_gradient_vector","operation":"gradient-rational-point","semantic_template":"qf-mod44-gradient-rational-point/{\"dimension\":2,\"point\":[[7,1],[4,1]],\"terms\":[[[1,1],[0,0]],[[7,1],[2,1]],[[8,1],[0,1]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r06-gradient-vector-0033","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-rational-point；資料={\"dimension\":3,\"terms\":[[8,[2,1,0]],[9,[0,1,1]],[1,[0,0,0]]],\"point\":[[8,1],[9,2],[10,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, [[1, [0, 0, 0]], [9, [0, 1, 1]], [8, [2, 1, 0]]], [8, 9/2, 10/3], [[[16, [1, 1, 0]]], [[9, [0, 0, 1]], [8, [2, 0, 0]]], [[9, [0, 1, 0]]]], [576, 542, 81/2], exact_rational_coordinate_evaluation)","explanation_zh":"Wave6第七層核對=(3, [[1, [0, 0, 0]], [9, [0, 1, 1]], [8, [2, 1, 0]]], [8, 9/2, 10/3], [[[16, [1, 1, 0]]], [[9, [0, 0, 1]], [8, [2, 0, 0]]], [[9, [0, 1, 0]]]], [576, 542, 81/2], exact_rational_coordinate_evaluation)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod44-gradient-rational-point","seed":202611136033,"parameters":{"dimension":3,"terms":[[8,[2,1,0]],[9,[0,1,1]],[1,[0,0,0]]],"point":[[8,1],[9,2],[10,3]],"domain":"Exact_polynomial_gradient_vector","operation":"gradient-rational-point","semantic_template":"qf-mod44-gradient-rational-point/{\"dimension\":3,\"point\":[[8,1],[9,2],[10,3]],\"terms\":[[[1,1],[0,0,0]],[[8,1],[2,1,0]],[[9,1],[0,1,1]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r06-gradient-vector-0034","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-rational-point；資料={\"dimension\":2,\"terms\":[[9,[2,1]],[10,[0,1]],[1,[0,0]]],\"point\":[[9,1],[10,2]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [[1, [0, 0]], [10, [0, 1]], [9, [2, 1]]], [9, 5], [[[18, [1, 1]]], [[10, [0, 0]], [9, [2, 0]]]], [810, 739], exact_rational_coordinate_evaluation)","explanation_zh":"Wave6第七層核對=(2, [[1, [0, 0]], [10, [0, 1]], [9, [2, 1]]], [9, 5], [[[18, [1, 1]]], [[10, [0, 0]], [9, [2, 0]]]], [810, 739], exact_rational_coordinate_evaluation)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod44-gradient-rational-point","seed":202611136034,"parameters":{"dimension":2,"terms":[[9,[2,1]],[10,[0,1]],[1,[0,0]]],"point":[[9,1],[10,2]],"domain":"Exact_polynomial_gradient_vector","operation":"gradient-rational-point","semantic_template":"qf-mod44-gradient-rational-point/{\"dimension\":2,\"point\":[[9,1],[5,1]],\"terms\":[[[1,1],[0,0]],[[10,1],[0,1]],[[9,1],[2,1]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r06-gradient-vector-0035","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-rational-point；資料={\"dimension\":3,\"terms\":[[10,[2,1,0]],[11,[0,1,1]],[1,[0,0,0]]],\"point\":[[10,1],[11,2],[12,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, [[1, [0, 0, 0]], [11, [0, 1, 1]], [10, [2, 1, 0]]], [10, 11/2, 4], [[[20, [1, 1, 0]]], [[11, [0, 0, 1]], [10, [2, 0, 0]]], [[11, [0, 1, 0]]]], [1100, 1044, 121/2], exact_rational_coordinate_evaluation)","explanation_zh":"Wave6第七層核對=(3, [[1, [0, 0, 0]], [11, [0, 1, 1]], [10, [2, 1, 0]]], [10, 11/2, 4], [[[20, [1, 1, 0]]], [[11, [0, 0, 1]], [10, [2, 0, 0]]], [[11, [0, 1, 0]]]], [1100, 1044, 121/2], exact_rational_coordinate_evaluation)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod44-gradient-rational-point","seed":202611136035,"parameters":{"dimension":3,"terms":[[10,[2,1,0]],[11,[0,1,1]],[1,[0,0,0]]],"point":[[10,1],[11,2],[12,3]],"domain":"Exact_polynomial_gradient_vector","operation":"gradient-rational-point","semantic_template":"qf-mod44-gradient-rational-point/{\"dimension\":3,\"point\":[[10,1],[11,2],[4,1]],\"terms\":[[[1,1],[0,0,0]],[[10,1],[2,1,0]],[[11,1],[0,1,1]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r06-gradient-vector-0036","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-rational-point；資料={\"dimension\":2,\"terms\":[[11,[2,1]],[12,[0,1]],[1,[0,0]]],\"point\":[[11,1],[12,2]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [[1, [0, 0]], [12, [0, 1]], [11, [2, 1]]], [11, 6], [[[22, [1, 1]]], [[12, [0, 0]], [11, [2, 0]]]], [1452, 1343], exact_rational_coordinate_evaluation)","explanation_zh":"Wave6第七層核對=(2, [[1, [0, 0]], [12, [0, 1]], [11, [2, 1]]], [11, 6], [[[22, [1, 1]]], [[12, [0, 0]], [11, [2, 0]]]], [1452, 1343], exact_rational_coordinate_evaluation)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod44-gradient-rational-point","seed":202611136036,"parameters":{"dimension":2,"terms":[[11,[2,1]],[12,[0,1]],[1,[0,0]]],"point":[[11,1],[12,2]],"domain":"Exact_polynomial_gradient_vector","operation":"gradient-rational-point","semantic_template":"qf-mod44-gradient-rational-point/{\"dimension\":2,\"point\":[[11,1],[6,1]],\"terms\":[[[1,1],[0,0]],[[11,1],[2,1]],[[12,1],[0,1]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r06-gradient-vector-0037","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-rational-point；資料={\"dimension\":3,\"terms\":[[12,[2,1,0]],[13,[0,1,1]],[1,[0,0,0]]],\"point\":[[12,1],[13,2],[14,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, [[1, [0, 0, 0]], [13, [0, 1, 1]], [12, [2, 1, 0]]], [12, 13/2, 14/3], [[[24, [1, 1, 0]]], [[13, [0, 0, 1]], [12, [2, 0, 0]]], [[13, [0, 1, 0]]]], [1872, 5366/3, 169/2], exact_rational_coordinate_evaluation)","explanation_zh":"Wave6第七層核對=(3, [[1, [0, 0, 0]], [13, [0, 1, 1]], [12, [2, 1, 0]]], [12, 13/2, 14/3], [[[24, [1, 1, 0]]], [[13, [0, 0, 1]], [12, [2, 0, 0]]], [[13, [0, 1, 0]]]], [1872, 5366/3, 169/2], exact_rational_coordinate_evaluation)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod44-gradient-rational-point","seed":202611136037,"parameters":{"dimension":3,"terms":[[12,[2,1,0]],[13,[0,1,1]],[1,[0,0,0]]],"point":[[12,1],[13,2],[14,3]],"domain":"Exact_polynomial_gradient_vector","operation":"gradient-rational-point","semantic_template":"qf-mod44-gradient-rational-point/{\"dimension\":3,\"point\":[[12,1],[13,2],[14,3]],\"terms\":[[[1,1],[0,0,0]],[[12,1],[2,1,0]],[[13,1],[0,1,1]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r06-gradient-vector-0038","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-rational-point；資料={\"dimension\":2,\"terms\":[[13,[2,1]],[14,[0,1]],[1,[0,0]]],\"point\":[[13,1],[14,2]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [[1, [0, 0]], [14, [0, 1]], [13, [2, 1]]], [13, 7], [[[26, [1, 1]]], [[14, [0, 0]], [13, [2, 0]]]], [2366, 2211], exact_rational_coordinate_evaluation)","explanation_zh":"Wave6第七層核對=(2, [[1, [0, 0]], [14, [0, 1]], [13, [2, 1]]], [13, 7], [[[26, [1, 1]]], [[14, [0, 0]], [13, [2, 0]]]], [2366, 2211], exact_rational_coordinate_evaluation)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod44-gradient-rational-point","seed":202611136038,"parameters":{"dimension":2,"terms":[[13,[2,1]],[14,[0,1]],[1,[0,0]]],"point":[[13,1],[14,2]],"domain":"Exact_polynomial_gradient_vector","operation":"gradient-rational-point","semantic_template":"qf-mod44-gradient-rational-point/{\"dimension\":2,\"point\":[[13,1],[7,1]],\"terms\":[[[1,1],[0,0]],[[13,1],[2,1]],[[14,1],[0,1]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r06-gradient-vector-0039","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-rational-point；資料={\"dimension\":3,\"terms\":[[14,[2,1,0]],[15,[0,1,1]],[1,[0,0,0]]],\"point\":[[14,1],[15,2],[16,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, [[1, [0, 0, 0]], [15, [0, 1, 1]], [14, [2, 1, 0]]], [14, 15/2, 16/3], [[[28, [1, 1, 0]]], [[15, [0, 0, 1]], [14, [2, 0, 0]]], [[15, [0, 1, 0]]]], [2940, 2824, 225/2], exact_rational_coordinate_evaluation)","explanation_zh":"Wave6第七層核對=(3, [[1, [0, 0, 0]], [15, [0, 1, 1]], [14, [2, 1, 0]]], [14, 15/2, 16/3], [[[28, [1, 1, 0]]], [[15, [0, 0, 1]], [14, [2, 0, 0]]], [[15, [0, 1, 0]]]], [2940, 2824, 225/2], exact_rational_coordinate_evaluation)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod44-gradient-rational-point","seed":202611136039,"parameters":{"dimension":3,"terms":[[14,[2,1,0]],[15,[0,1,1]],[1,[0,0,0]]],"point":[[14,1],[15,2],[16,3]],"domain":"Exact_polynomial_gradient_vector","operation":"gradient-rational-point","semantic_template":"qf-mod44-gradient-rational-point/{\"dimension\":3,\"point\":[[14,1],[15,2],[16,3]],\"terms\":[[[1,1],[0,0,0]],[[14,1],[2,1,0]],[[15,1],[0,1,1]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r06-gradient-vector-0040","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-rational-point；資料={\"dimension\":2,\"terms\":[[15,[2,1]],[16,[0,1]],[1,[0,0]]],\"point\":[[15,1],[16,2]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [[1, [0, 0]], [16, [0, 1]], [15, [2, 1]]], [15, 8], [[[30, [1, 1]]], [[16, [0, 0]], [15, [2, 0]]]], [3600, 3391], exact_rational_coordinate_evaluation)","explanation_zh":"Wave6第七層核對=(2, [[1, [0, 0]], [16, [0, 1]], [15, [2, 1]]], [15, 8], [[[30, [1, 1]]], [[16, [0, 0]], [15, [2, 0]]]], [3600, 3391], exact_rational_coordinate_evaluation)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod44-gradient-rational-point","seed":202611136040,"parameters":{"dimension":2,"terms":[[15,[2,1]],[16,[0,1]],[1,[0,0]]],"point":[[15,1],[16,2]],"domain":"Exact_polynomial_gradient_vector","operation":"gradient-rational-point","semantic_template":"qf-mod44-gradient-rational-point/{\"dimension\":2,\"point\":[[15,1],[8,1]],\"terms\":[[[1,1],[0,0]],[[15,1],[2,1]],[[16,1],[0,1]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r06-gradient-vector-0041","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-rational-point；資料={\"dimension\":3,\"terms\":[[16,[2,1,0]],[17,[0,1,1]],[1,[0,0,0]]],\"point\":[[16,1],[17,2],[18,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, [[1, [0, 0, 0]], [17, [0, 1, 1]], [16, [2, 1, 0]]], [16, 17/2, 6], [[[32, [1, 1, 0]]], [[17, [0, 0, 1]], [16, [2, 0, 0]]], [[17, [0, 1, 0]]]], [4352, 4198, 289/2], exact_rational_coordinate_evaluation)","explanation_zh":"Wave6第七層核對=(3, [[1, [0, 0, 0]], [17, [0, 1, 1]], [16, [2, 1, 0]]], [16, 17/2, 6], [[[32, [1, 1, 0]]], [[17, [0, 0, 1]], [16, [2, 0, 0]]], [[17, [0, 1, 0]]]], [4352, 4198, 289/2], exact_rational_coordinate_evaluation)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod44-gradient-rational-point","seed":202611136041,"parameters":{"dimension":3,"terms":[[16,[2,1,0]],[17,[0,1,1]],[1,[0,0,0]]],"point":[[16,1],[17,2],[18,3]],"domain":"Exact_polynomial_gradient_vector","operation":"gradient-rational-point","semantic_template":"qf-mod44-gradient-rational-point/{\"dimension\":3,\"point\":[[16,1],[17,2],[6,1]],\"terms\":[[[1,1],[0,0,0]],[[16,1],[2,1,0]],[[17,1],[0,1,1]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r06-gradient-vector-0042","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-rational-point；資料={\"dimension\":2,\"terms\":[[17,[2,1]],[18,[0,1]],[1,[0,0]]],\"point\":[[17,1],[18,2]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [[1, [0, 0]], [18, [0, 1]], [17, [2, 1]]], [17, 9], [[[34, [1, 1]]], [[18, [0, 0]], [17, [2, 0]]]], [5202, 4931], exact_rational_coordinate_evaluation)","explanation_zh":"Wave6第七層核對=(2, [[1, [0, 0]], [18, [0, 1]], [17, [2, 1]]], [17, 9], [[[34, [1, 1]]], [[18, [0, 0]], [17, [2, 0]]]], [5202, 4931], exact_rational_coordinate_evaluation)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod44-gradient-rational-point","seed":202611136042,"parameters":{"dimension":2,"terms":[[17,[2,1]],[18,[0,1]],[1,[0,0]]],"point":[[17,1],[18,2]],"domain":"Exact_polynomial_gradient_vector","operation":"gradient-rational-point","semantic_template":"qf-mod44-gradient-rational-point/{\"dimension\":2,\"point\":[[17,1],[9,1]],\"terms\":[[[1,1],[0,0]],[[17,1],[2,1]],[[18,1],[0,1]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r06-gradient-vector-0043","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-rational-point；資料={\"dimension\":3,\"terms\":[[18,[2,1,0]],[19,[0,1,1]],[1,[0,0,0]]],\"point\":[[18,1],[19,2],[20,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, [[1, [0, 0, 0]], [19, [0, 1, 1]], [18, [2, 1, 0]]], [18, 19/2, 20/3], [[[36, [1, 1, 0]]], [[19, [0, 0, 1]], [18, [2, 0, 0]]], [[19, [0, 1, 0]]]], [6156, 17876/3, 361/2], exact_rational_coordinate_evaluation)","explanation_zh":"Wave6第七層核對=(3, [[1, [0, 0, 0]], [19, [0, 1, 1]], [18, [2, 1, 0]]], [18, 19/2, 20/3], [[[36, [1, 1, 0]]], [[19, [0, 0, 1]], [18, [2, 0, 0]]], [[19, [0, 1, 0]]]], [6156, 17876/3, 361/2], exact_rational_coordinate_evaluation)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod44-gradient-rational-point","seed":202611136043,"parameters":{"dimension":3,"terms":[[18,[2,1,0]],[19,[0,1,1]],[1,[0,0,0]]],"point":[[18,1],[19,2],[20,3]],"domain":"Exact_polynomial_gradient_vector","operation":"gradient-rational-point","semantic_template":"qf-mod44-gradient-rational-point/{\"dimension\":3,\"point\":[[18,1],[19,2],[20,3]],\"terms\":[[[1,1],[0,0,0]],[[18,1],[2,1,0]],[[19,1],[0,1,1]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r06-gradient-vector-0044","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-rational-point；資料={\"dimension\":2,\"terms\":[[19,[2,1]],[20,[0,1]],[1,[0,0]]],\"point\":[[19,1],[20,2]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [[1, [0, 0]], [20, [0, 1]], [19, [2, 1]]], [19, 10], [[[38, [1, 1]]], [[20, [0, 0]], [19, [2, 0]]]], [7220, 6879], exact_rational_coordinate_evaluation)","explanation_zh":"Wave6第七層核對=(2, [[1, [0, 0]], [20, [0, 1]], [19, [2, 1]]], [19, 10], [[[38, [1, 1]]], [[20, [0, 0]], [19, [2, 0]]]], [7220, 6879], exact_rational_coordinate_evaluation)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod44-gradient-rational-point","seed":202611136044,"parameters":{"dimension":2,"terms":[[19,[2,1]],[20,[0,1]],[1,[0,0]]],"point":[[19,1],[20,2]],"domain":"Exact_polynomial_gradient_vector","operation":"gradient-rational-point","semantic_template":"qf-mod44-gradient-rational-point/{\"dimension\":2,\"point\":[[19,1],[10,1]],\"terms\":[[[1,1],[0,0]],[[19,1],[2,1]],[[20,1],[0,1]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r06-gradient-vector-0045","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-rational-point；資料={\"dimension\":3,\"terms\":[[20,[2,1,0]],[21,[0,1,1]],[1,[0,0,0]]],\"point\":[[20,1],[21,2],[22,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, [[1, [0, 0, 0]], [21, [0, 1, 1]], [20, [2, 1, 0]]], [20, 21/2, 22/3], [[[40, [1, 1, 0]]], [[21, [0, 0, 1]], [20, [2, 0, 0]]], [[21, [0, 1, 0]]]], [8400, 8154, 441/2], exact_rational_coordinate_evaluation)","explanation_zh":"Wave6第七層核對=(3, [[1, [0, 0, 0]], [21, [0, 1, 1]], [20, [2, 1, 0]]], [20, 21/2, 22/3], [[[40, [1, 1, 0]]], [[21, [0, 0, 1]], [20, [2, 0, 0]]], [[21, [0, 1, 0]]]], [8400, 8154, 441/2], exact_rational_coordinate_evaluation)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod44-gradient-rational-point","seed":202611136045,"parameters":{"dimension":3,"terms":[[20,[2,1,0]],[21,[0,1,1]],[1,[0,0,0]]],"point":[[20,1],[21,2],[22,3]],"domain":"Exact_polynomial_gradient_vector","operation":"gradient-rational-point","semantic_template":"qf-mod44-gradient-rational-point/{\"dimension\":3,\"point\":[[20,1],[21,2],[22,3]],\"terms\":[[[1,1],[0,0,0]],[[20,1],[2,1,0]],[[21,1],[0,1,1]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r06-gradient-vector-0046","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-rational-point；資料={\"dimension\":2,\"terms\":[[21,[2,1]],[22,[0,1]],[1,[0,0]]],\"point\":[[21,1],[22,2]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [[1, [0, 0]], [22, [0, 1]], [21, [2, 1]]], [21, 11], [[[42, [1, 1]]], [[22, [0, 0]], [21, [2, 0]]]], [9702, 9283], exact_rational_coordinate_evaluation)","explanation_zh":"Wave6第七層核對=(2, [[1, [0, 0]], [22, [0, 1]], [21, [2, 1]]], [21, 11], [[[42, [1, 1]]], [[22, [0, 0]], [21, [2, 0]]]], [9702, 9283], exact_rational_coordinate_evaluation)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod44-gradient-rational-point","seed":202611136046,"parameters":{"dimension":2,"terms":[[21,[2,1]],[22,[0,1]],[1,[0,0]]],"point":[[21,1],[22,2]],"domain":"Exact_polynomial_gradient_vector","operation":"gradient-rational-point","semantic_template":"qf-mod44-gradient-rational-point/{\"dimension\":2,\"point\":[[21,1],[11,1]],\"terms\":[[[1,1],[0,0]],[[21,1],[2,1]],[[22,1],[0,1]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r06-gradient-vector-0047","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-rational-point；資料={\"dimension\":3,\"terms\":[[22,[2,1,0]],[23,[0,1,1]],[1,[0,0,0]]],\"point\":[[22,1],[23,2],[24,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, [[1, [0, 0, 0]], [23, [0, 1, 1]], [22, [2, 1, 0]]], [22, 23/2, 8], [[[44, [1, 1, 0]]], [[23, [0, 0, 1]], [22, [2, 0, 0]]], [[23, [0, 1, 0]]]], [11132, 10832, 529/2], exact_rational_coordinate_evaluation)","explanation_zh":"Wave6第七層核對=(3, [[1, [0, 0, 0]], [23, [0, 1, 1]], [22, [2, 1, 0]]], [22, 23/2, 8], [[[44, [1, 1, 0]]], [[23, [0, 0, 1]], [22, [2, 0, 0]]], [[23, [0, 1, 0]]]], [11132, 10832, 529/2], exact_rational_coordinate_evaluation)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod44-gradient-rational-point","seed":202611136047,"parameters":{"dimension":3,"terms":[[22,[2,1,0]],[23,[0,1,1]],[1,[0,0,0]]],"point":[[22,1],[23,2],[24,3]],"domain":"Exact_polynomial_gradient_vector","operation":"gradient-rational-point","semantic_template":"qf-mod44-gradient-rational-point/{\"dimension\":3,\"point\":[[22,1],[23,2],[8,1]],\"terms\":[[[1,1],[0,0,0]],[[22,1],[2,1,0]],[[23,1],[0,1,1]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r06-gradient-vector-0048","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-rational-point；資料={\"dimension\":2,\"terms\":[[23,[2,1]],[24,[0,1]],[1,[0,0]]],\"point\":[[23,1],[24,2]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [[1, [0, 0]], [24, [0, 1]], [23, [2, 1]]], [23, 12], [[[46, [1, 1]]], [[24, [0, 0]], [23, [2, 0]]]], [12696, 12191], exact_rational_coordinate_evaluation)","explanation_zh":"Wave6第七層核對=(2, [[1, [0, 0]], [24, [0, 1]], [23, [2, 1]]], [23, 12], [[[46, [1, 1]]], [[24, [0, 0]], [23, [2, 0]]]], [12696, 12191], exact_rational_coordinate_evaluation)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod44-gradient-rational-point","seed":202611136048,"parameters":{"dimension":2,"terms":[[23,[2,1]],[24,[0,1]],[1,[0,0]]],"point":[[23,1],[24,2]],"domain":"Exact_polynomial_gradient_vector","operation":"gradient-rational-point","semantic_template":"qf-mod44-gradient-rational-point/{\"dimension\":2,\"point\":[[23,1],[12,1]],\"terms\":[[[1,1],[0,0]],[[23,1],[2,1]],[[24,1],[0,1]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r06-gradient-vector-0049","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-rational-point；資料={\"dimension\":3,\"terms\":[[24,[2,1,0]],[25,[0,1,1]],[1,[0,0,0]]],\"point\":[[24,1],[25,2],[26,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, [[1, [0, 0, 0]], [25, [0, 1, 1]], [24, [2, 1, 0]]], [24, 25/2, 26/3], [[[48, [1, 1, 0]]], [[25, [0, 0, 1]], [24, [2, 0, 0]]], [[25, [0, 1, 0]]]], [14400, 42122/3, 625/2], exact_rational_coordinate_evaluation)","explanation_zh":"Wave6第七層核對=(3, [[1, [0, 0, 0]], [25, [0, 1, 1]], [24, [2, 1, 0]]], [24, 25/2, 26/3], [[[48, [1, 1, 0]]], [[25, [0, 0, 1]], [24, [2, 0, 0]]], [[25, [0, 1, 0]]]], [14400, 42122/3, 625/2], exact_rational_coordinate_evaluation)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod44-gradient-rational-point","seed":202611136049,"parameters":{"dimension":3,"terms":[[24,[2,1,0]],[25,[0,1,1]],[1,[0,0,0]]],"point":[[24,1],[25,2],[26,3]],"domain":"Exact_polynomial_gradient_vector","operation":"gradient-rational-point","semantic_template":"qf-mod44-gradient-rational-point/{\"dimension\":3,\"point\":[[24,1],[25,2],[26,3]],\"terms\":[[[1,1],[0,0,0]],[[24,1],[2,1,0]],[[25,1],[0,1,1]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r06-gradient-vector-0050","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-rational-point；資料={\"dimension\":2,\"terms\":[[25,[2,1]],[26,[0,1]],[1,[0,0]]],\"point\":[[25,1],[26,2]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [[1, [0, 0]], [26, [0, 1]], [25, [2, 1]]], [25, 13], [[[50, [1, 1]]], [[26, [0, 0]], [25, [2, 0]]]], [16250, 15651], exact_rational_coordinate_evaluation)","explanation_zh":"Wave6第七層核對=(2, [[1, [0, 0]], [26, [0, 1]], [25, [2, 1]]], [25, 13], [[[50, [1, 1]]], [[26, [0, 0]], [25, [2, 0]]]], [16250, 15651], exact_rational_coordinate_evaluation)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod44-gradient-rational-point","seed":202611136050,"parameters":{"dimension":2,"terms":[[25,[2,1]],[26,[0,1]],[1,[0,0]]],"point":[[25,1],[26,2]],"domain":"Exact_polynomial_gradient_vector","operation":"gradient-rational-point","semantic_template":"qf-mod44-gradient-rational-point/{\"dimension\":2,\"point\":[[25,1],[13,1]],\"terms\":[[[1,1],[0,0]],[[25,1],[2,1]],[[26,1],[0,1]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r06-gradient-vector-0051","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-stationary-audit；資料={\"dimension\":2,\"terms\":[[1,[2,0]],[2,[0,2]]],\"point\":[0,0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [0, 0], [0, 0], 是, 否, gradient_zero_means_stationary_not_necessarily_extremum)","explanation_zh":"Wave6第七層核對=(2, [0, 0], [0, 0], 是, 否, gradient_zero_means_stationary_not_necessarily_extremum)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod44-gradient-stationary-audit","seed":202611136051,"parameters":{"dimension":2,"terms":[[1,[2,0]],[2,[0,2]]],"point":[0,0],"domain":"Exact_polynomial_gradient_vector","operation":"gradient-stationary-audit","semantic_template":"qf-mod44-gradient-stationary-audit/{\"dimension\":2,\"point\":[[0,1],[0,1]],\"terms\":[[[1,1],[2,0]],[[2,1],[0,2]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r06-gradient-vector-0052","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-stationary-audit；資料={\"dimension\":2,\"terms\":[[2,[2,0]],[3,[0,2]]],\"point\":[0,0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [0, 0], [0, 0], 是, 否, gradient_zero_means_stationary_not_necessarily_extremum)","explanation_zh":"Wave6第七層核對=(2, [0, 0], [0, 0], 是, 否, gradient_zero_means_stationary_not_necessarily_extremum)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod44-gradient-stationary-audit","seed":202611136052,"parameters":{"dimension":2,"terms":[[2,[2,0]],[3,[0,2]]],"point":[0,0],"domain":"Exact_polynomial_gradient_vector","operation":"gradient-stationary-audit","semantic_template":"qf-mod44-gradient-stationary-audit/{\"dimension\":2,\"point\":[[0,1],[0,1]],\"terms\":[[[2,1],[2,0]],[[3,1],[0,2]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r06-gradient-vector-0053","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-stationary-audit；資料={\"dimension\":2,\"terms\":[[3,[2,0]],[4,[0,2]]],\"point\":[0,0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [0, 0], [0, 0], 是, 否, gradient_zero_means_stationary_not_necessarily_extremum)","explanation_zh":"Wave6第七層核對=(2, [0, 0], [0, 0], 是, 否, gradient_zero_means_stationary_not_necessarily_extremum)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod44-gradient-stationary-audit","seed":202611136053,"parameters":{"dimension":2,"terms":[[3,[2,0]],[4,[0,2]]],"point":[0,0],"domain":"Exact_polynomial_gradient_vector","operation":"gradient-stationary-audit","semantic_template":"qf-mod44-gradient-stationary-audit/{\"dimension\":2,\"point\":[[0,1],[0,1]],\"terms\":[[[3,1],[2,0]],[[4,1],[0,2]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r06-gradient-vector-0054","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-stationary-audit；資料={\"dimension\":2,\"terms\":[[4,[2,0]],[5,[0,2]]],\"point\":[0,0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [0, 0], [0, 0], 是, 否, gradient_zero_means_stationary_not_necessarily_extremum)","explanation_zh":"Wave6第七層核對=(2, [0, 0], [0, 0], 是, 否, gradient_zero_means_stationary_not_necessarily_extremum)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod44-gradient-stationary-audit","seed":202611136054,"parameters":{"dimension":2,"terms":[[4,[2,0]],[5,[0,2]]],"point":[0,0],"domain":"Exact_polynomial_gradient_vector","operation":"gradient-stationary-audit","semantic_template":"qf-mod44-gradient-stationary-audit/{\"dimension\":2,\"point\":[[0,1],[0,1]],\"terms\":[[[4,1],[2,0]],[[5,1],[0,2]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r06-gradient-vector-0055","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-stationary-audit；資料={\"dimension\":2,\"terms\":[[5,[2,0]],[6,[0,2]]],\"point\":[0,0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [0, 0], [0, 0], 是, 否, gradient_zero_means_stationary_not_necessarily_extremum)","explanation_zh":"Wave6第七層核對=(2, [0, 0], [0, 0], 是, 否, gradient_zero_means_stationary_not_necessarily_extremum)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod44-gradient-stationary-audit","seed":202611136055,"parameters":{"dimension":2,"terms":[[5,[2,0]],[6,[0,2]]],"point":[0,0],"domain":"Exact_polynomial_gradient_vector","operation":"gradient-stationary-audit","semantic_template":"qf-mod44-gradient-stationary-audit/{\"dimension\":2,\"point\":[[0,1],[0,1]],\"terms\":[[[5,1],[2,0]],[[6,1],[0,2]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r06-gradient-vector-0056","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-stationary-audit；資料={\"dimension\":2,\"terms\":[[6,[2,0]],[7,[0,2]]],\"point\":[0,0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [0, 0], [0, 0], 是, 否, gradient_zero_means_stationary_not_necessarily_extremum)","explanation_zh":"Wave6第七層核對=(2, [0, 0], [0, 0], 是, 否, gradient_zero_means_stationary_not_necessarily_extremum)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod44-gradient-stationary-audit","seed":202611136056,"parameters":{"dimension":2,"terms":[[6,[2,0]],[7,[0,2]]],"point":[0,0],"domain":"Exact_polynomial_gradient_vector","operation":"gradient-stationary-audit","semantic_template":"qf-mod44-gradient-stationary-audit/{\"dimension\":2,\"point\":[[0,1],[0,1]],\"terms\":[[[6,1],[2,0]],[[7,1],[0,2]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r06-gradient-vector-0057","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-stationary-audit；資料={\"dimension\":2,\"terms\":[[7,[2,0]],[8,[0,2]]],\"point\":[0,0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [0, 0], [0, 0], 是, 否, gradient_zero_means_stationary_not_necessarily_extremum)","explanation_zh":"Wave6第七層核對=(2, [0, 0], [0, 0], 是, 否, gradient_zero_means_stationary_not_necessarily_extremum)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod44-gradient-stationary-audit","seed":202611136057,"parameters":{"dimension":2,"terms":[[7,[2,0]],[8,[0,2]]],"point":[0,0],"domain":"Exact_polynomial_gradient_vector","operation":"gradient-stationary-audit","semantic_template":"qf-mod44-gradient-stationary-audit/{\"dimension\":2,\"point\":[[0,1],[0,1]],\"terms\":[[[7,1],[2,0]],[[8,1],[0,2]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r06-gradient-vector-0058","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-stationary-audit；資料={\"dimension\":2,\"terms\":[[8,[2,0]],[9,[0,2]]],\"point\":[0,0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [0, 0], [0, 0], 是, 否, gradient_zero_means_stationary_not_necessarily_extremum)","explanation_zh":"Wave6第七層核對=(2, [0, 0], [0, 0], 是, 否, gradient_zero_means_stationary_not_necessarily_extremum)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod44-gradient-stationary-audit","seed":202611136058,"parameters":{"dimension":2,"terms":[[8,[2,0]],[9,[0,2]]],"point":[0,0],"domain":"Exact_polynomial_gradient_vector","operation":"gradient-stationary-audit","semantic_template":"qf-mod44-gradient-stationary-audit/{\"dimension\":2,\"point\":[[0,1],[0,1]],\"terms\":[[[8,1],[2,0]],[[9,1],[0,2]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r06-gradient-vector-0059","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-stationary-audit；資料={\"dimension\":2,\"terms\":[[9,[2,0]],[10,[0,2]]],\"point\":[0,0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [0, 0], [0, 0], 是, 否, gradient_zero_means_stationary_not_necessarily_extremum)","explanation_zh":"Wave6第七層核對=(2, [0, 0], [0, 0], 是, 否, gradient_zero_means_stationary_not_necessarily_extremum)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod44-gradient-stationary-audit","seed":202611136059,"parameters":{"dimension":2,"terms":[[9,[2,0]],[10,[0,2]]],"point":[0,0],"domain":"Exact_polynomial_gradient_vector","operation":"gradient-stationary-audit","semantic_template":"qf-mod44-gradient-stationary-audit/{\"dimension\":2,\"point\":[[0,1],[0,1]],\"terms\":[[[10,1],[0,2]],[[9,1],[2,0]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r06-gradient-vector-0060","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-stationary-audit；資料={\"dimension\":2,\"terms\":[[10,[2,0]],[11,[0,2]]],\"point\":[0,0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [0, 0], [0, 0], 是, 否, gradient_zero_means_stationary_not_necessarily_extremum)","explanation_zh":"Wave6第七層核對=(2, [0, 0], [0, 0], 是, 否, gradient_zero_means_stationary_not_necessarily_extremum)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod44-gradient-stationary-audit","seed":202611136060,"parameters":{"dimension":2,"terms":[[10,[2,0]],[11,[0,2]]],"point":[0,0],"domain":"Exact_polynomial_gradient_vector","operation":"gradient-stationary-audit","semantic_template":"qf-mod44-gradient-stationary-audit/{\"dimension\":2,\"point\":[[0,1],[0,1]],\"terms\":[[[10,1],[2,0]],[[11,1],[0,2]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r06-gradient-vector-0061","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-stationary-audit；資料={\"dimension\":2,\"terms\":[[11,[2,0]],[12,[0,2]]],\"point\":[0,0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [0, 0], [0, 0], 是, 否, gradient_zero_means_stationary_not_necessarily_extremum)","explanation_zh":"Wave6第七層核對=(2, [0, 0], [0, 0], 是, 否, gradient_zero_means_stationary_not_necessarily_extremum)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod44-gradient-stationary-audit","seed":202611136061,"parameters":{"dimension":2,"terms":[[11,[2,0]],[12,[0,2]]],"point":[0,0],"domain":"Exact_polynomial_gradient_vector","operation":"gradient-stationary-audit","semantic_template":"qf-mod44-gradient-stationary-audit/{\"dimension\":2,\"point\":[[0,1],[0,1]],\"terms\":[[[11,1],[2,0]],[[12,1],[0,2]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r06-gradient-vector-0062","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-stationary-audit；資料={\"dimension\":2,\"terms\":[[12,[2,0]],[13,[0,2]]],\"point\":[0,0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [0, 0], [0, 0], 是, 否, gradient_zero_means_stationary_not_necessarily_extremum)","explanation_zh":"Wave6第七層核對=(2, [0, 0], [0, 0], 是, 否, gradient_zero_means_stationary_not_necessarily_extremum)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod44-gradient-stationary-audit","seed":202611136062,"parameters":{"dimension":2,"terms":[[12,[2,0]],[13,[0,2]]],"point":[0,0],"domain":"Exact_polynomial_gradient_vector","operation":"gradient-stationary-audit","semantic_template":"qf-mod44-gradient-stationary-audit/{\"dimension\":2,\"point\":[[0,1],[0,1]],\"terms\":[[[12,1],[2,0]],[[13,1],[0,2]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r06-gradient-vector-0063","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-stationary-audit；資料={\"dimension\":2,\"terms\":[[13,[2,0]],[14,[0,2]]],\"point\":[0,0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [0, 0], [0, 0], 是, 否, gradient_zero_means_stationary_not_necessarily_extremum)","explanation_zh":"Wave6第七層核對=(2, [0, 0], [0, 0], 是, 否, gradient_zero_means_stationary_not_necessarily_extremum)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod44-gradient-stationary-audit","seed":202611136063,"parameters":{"dimension":2,"terms":[[13,[2,0]],[14,[0,2]]],"point":[0,0],"domain":"Exact_polynomial_gradient_vector","operation":"gradient-stationary-audit","semantic_template":"qf-mod44-gradient-stationary-audit/{\"dimension\":2,\"point\":[[0,1],[0,1]],\"terms\":[[[13,1],[2,0]],[[14,1],[0,2]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r06-gradient-vector-0064","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-stationary-audit；資料={\"dimension\":2,\"terms\":[[14,[2,0]],[15,[0,2]]],\"point\":[0,0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [0, 0], [0, 0], 是, 否, gradient_zero_means_stationary_not_necessarily_extremum)","explanation_zh":"Wave6第七層核對=(2, [0, 0], [0, 0], 是, 否, gradient_zero_means_stationary_not_necessarily_extremum)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod44-gradient-stationary-audit","seed":202611136064,"parameters":{"dimension":2,"terms":[[14,[2,0]],[15,[0,2]]],"point":[0,0],"domain":"Exact_polynomial_gradient_vector","operation":"gradient-stationary-audit","semantic_template":"qf-mod44-gradient-stationary-audit/{\"dimension\":2,\"point\":[[0,1],[0,1]],\"terms\":[[[14,1],[2,0]],[[15,1],[0,2]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r06-gradient-vector-0065","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-stationary-audit；資料={\"dimension\":2,\"terms\":[[15,[2,0]],[16,[0,2]]],\"point\":[0,0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [0, 0], [0, 0], 是, 否, gradient_zero_means_stationary_not_necessarily_extremum)","explanation_zh":"Wave6第七層核對=(2, [0, 0], [0, 0], 是, 否, gradient_zero_means_stationary_not_necessarily_extremum)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod44-gradient-stationary-audit","seed":202611136065,"parameters":{"dimension":2,"terms":[[15,[2,0]],[16,[0,2]]],"point":[0,0],"domain":"Exact_polynomial_gradient_vector","operation":"gradient-stationary-audit","semantic_template":"qf-mod44-gradient-stationary-audit/{\"dimension\":2,\"point\":[[0,1],[0,1]],\"terms\":[[[15,1],[2,0]],[[16,1],[0,2]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r06-gradient-vector-0066","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-stationary-audit；資料={\"dimension\":2,\"terms\":[[16,[2,0]],[17,[0,2]]],\"point\":[0,0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [0, 0], [0, 0], 是, 否, gradient_zero_means_stationary_not_necessarily_extremum)","explanation_zh":"Wave6第七層核對=(2, [0, 0], [0, 0], 是, 否, gradient_zero_means_stationary_not_necessarily_extremum)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod44-gradient-stationary-audit","seed":202611136066,"parameters":{"dimension":2,"terms":[[16,[2,0]],[17,[0,2]]],"point":[0,0],"domain":"Exact_polynomial_gradient_vector","operation":"gradient-stationary-audit","semantic_template":"qf-mod44-gradient-stationary-audit/{\"dimension\":2,\"point\":[[0,1],[0,1]],\"terms\":[[[16,1],[2,0]],[[17,1],[0,2]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r06-gradient-vector-0067","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-stationary-audit；資料={\"dimension\":2,\"terms\":[[17,[2,0]],[18,[0,2]]],\"point\":[0,0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [0, 0], [0, 0], 是, 否, gradient_zero_means_stationary_not_necessarily_extremum)","explanation_zh":"Wave6第七層核對=(2, [0, 0], [0, 0], 是, 否, gradient_zero_means_stationary_not_necessarily_extremum)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod44-gradient-stationary-audit","seed":202611136067,"parameters":{"dimension":2,"terms":[[17,[2,0]],[18,[0,2]]],"point":[0,0],"domain":"Exact_polynomial_gradient_vector","operation":"gradient-stationary-audit","semantic_template":"qf-mod44-gradient-stationary-audit/{\"dimension\":2,\"point\":[[0,1],[0,1]],\"terms\":[[[17,1],[2,0]],[[18,1],[0,2]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r06-gradient-vector-0068","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-stationary-audit；資料={\"dimension\":2,\"terms\":[[18,[2,0]],[19,[0,2]]],\"point\":[0,0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [0, 0], [0, 0], 是, 否, gradient_zero_means_stationary_not_necessarily_extremum)","explanation_zh":"Wave6第七層核對=(2, [0, 0], [0, 0], 是, 否, gradient_zero_means_stationary_not_necessarily_extremum)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod44-gradient-stationary-audit","seed":202611136068,"parameters":{"dimension":2,"terms":[[18,[2,0]],[19,[0,2]]],"point":[0,0],"domain":"Exact_polynomial_gradient_vector","operation":"gradient-stationary-audit","semantic_template":"qf-mod44-gradient-stationary-audit/{\"dimension\":2,\"point\":[[0,1],[0,1]],\"terms\":[[[18,1],[2,0]],[[19,1],[0,2]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r06-gradient-vector-0069","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-stationary-audit；資料={\"dimension\":2,\"terms\":[[19,[2,0]],[20,[0,2]]],\"point\":[0,0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [0, 0], [0, 0], 是, 否, gradient_zero_means_stationary_not_necessarily_extremum)","explanation_zh":"Wave6第七層核對=(2, [0, 0], [0, 0], 是, 否, gradient_zero_means_stationary_not_necessarily_extremum)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod44-gradient-stationary-audit","seed":202611136069,"parameters":{"dimension":2,"terms":[[19,[2,0]],[20,[0,2]]],"point":[0,0],"domain":"Exact_polynomial_gradient_vector","operation":"gradient-stationary-audit","semantic_template":"qf-mod44-gradient-stationary-audit/{\"dimension\":2,\"point\":[[0,1],[0,1]],\"terms\":[[[19,1],[2,0]],[[20,1],[0,2]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r06-gradient-vector-0070","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-stationary-audit；資料={\"dimension\":2,\"terms\":[[20,[2,0]],[21,[0,2]]],\"point\":[0,0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [0, 0], [0, 0], 是, 否, gradient_zero_means_stationary_not_necessarily_extremum)","explanation_zh":"Wave6第七層核對=(2, [0, 0], [0, 0], 是, 否, gradient_zero_means_stationary_not_necessarily_extremum)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod44-gradient-stationary-audit","seed":202611136070,"parameters":{"dimension":2,"terms":[[20,[2,0]],[21,[0,2]]],"point":[0,0],"domain":"Exact_polynomial_gradient_vector","operation":"gradient-stationary-audit","semantic_template":"qf-mod44-gradient-stationary-audit/{\"dimension\":2,\"point\":[[0,1],[0,1]],\"terms\":[[[20,1],[2,0]],[[21,1],[0,2]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r06-gradient-vector-0071","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-stationary-audit；資料={\"dimension\":2,\"terms\":[[21,[2,0]],[22,[0,2]]],\"point\":[0,0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [0, 0], [0, 0], 是, 否, gradient_zero_means_stationary_not_necessarily_extremum)","explanation_zh":"Wave6第七層核對=(2, [0, 0], [0, 0], 是, 否, gradient_zero_means_stationary_not_necessarily_extremum)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod44-gradient-stationary-audit","seed":202611136071,"parameters":{"dimension":2,"terms":[[21,[2,0]],[22,[0,2]]],"point":[0,0],"domain":"Exact_polynomial_gradient_vector","operation":"gradient-stationary-audit","semantic_template":"qf-mod44-gradient-stationary-audit/{\"dimension\":2,\"point\":[[0,1],[0,1]],\"terms\":[[[21,1],[2,0]],[[22,1],[0,2]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r06-gradient-vector-0072","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-stationary-audit；資料={\"dimension\":2,\"terms\":[[22,[2,0]],[23,[0,2]]],\"point\":[0,0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [0, 0], [0, 0], 是, 否, gradient_zero_means_stationary_not_necessarily_extremum)","explanation_zh":"Wave6第七層核對=(2, [0, 0], [0, 0], 是, 否, gradient_zero_means_stationary_not_necessarily_extremum)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod44-gradient-stationary-audit","seed":202611136072,"parameters":{"dimension":2,"terms":[[22,[2,0]],[23,[0,2]]],"point":[0,0],"domain":"Exact_polynomial_gradient_vector","operation":"gradient-stationary-audit","semantic_template":"qf-mod44-gradient-stationary-audit/{\"dimension\":2,\"point\":[[0,1],[0,1]],\"terms\":[[[22,1],[2,0]],[[23,1],[0,2]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r06-gradient-vector-0073","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-stationary-audit；資料={\"dimension\":2,\"terms\":[[23,[2,0]],[24,[0,2]]],\"point\":[0,0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [0, 0], [0, 0], 是, 否, gradient_zero_means_stationary_not_necessarily_extremum)","explanation_zh":"Wave6第七層核對=(2, [0, 0], [0, 0], 是, 否, gradient_zero_means_stationary_not_necessarily_extremum)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod44-gradient-stationary-audit","seed":202611136073,"parameters":{"dimension":2,"terms":[[23,[2,0]],[24,[0,2]]],"point":[0,0],"domain":"Exact_polynomial_gradient_vector","operation":"gradient-stationary-audit","semantic_template":"qf-mod44-gradient-stationary-audit/{\"dimension\":2,\"point\":[[0,1],[0,1]],\"terms\":[[[23,1],[2,0]],[[24,1],[0,2]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r06-gradient-vector-0074","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-stationary-audit；資料={\"dimension\":2,\"terms\":[[24,[2,0]],[25,[0,2]]],\"point\":[0,0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [0, 0], [0, 0], 是, 否, gradient_zero_means_stationary_not_necessarily_extremum)","explanation_zh":"Wave6第七層核對=(2, [0, 0], [0, 0], 是, 否, gradient_zero_means_stationary_not_necessarily_extremum)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod44-gradient-stationary-audit","seed":202611136074,"parameters":{"dimension":2,"terms":[[24,[2,0]],[25,[0,2]]],"point":[0,0],"domain":"Exact_polynomial_gradient_vector","operation":"gradient-stationary-audit","semantic_template":"qf-mod44-gradient-stationary-audit/{\"dimension\":2,\"point\":[[0,1],[0,1]],\"terms\":[[[24,1],[2,0]],[[25,1],[0,2]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r06-gradient-vector-0075","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-stationary-audit；資料={\"dimension\":2,\"terms\":[[25,[2,0]],[26,[0,2]]],\"point\":[0,0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [0, 0], [0, 0], 是, 否, gradient_zero_means_stationary_not_necessarily_extremum)","explanation_zh":"Wave6第七層核對=(2, [0, 0], [0, 0], 是, 否, gradient_zero_means_stationary_not_necessarily_extremum)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod44-gradient-stationary-audit","seed":202611136075,"parameters":{"dimension":2,"terms":[[25,[2,0]],[26,[0,2]]],"point":[0,0],"domain":"Exact_polynomial_gradient_vector","operation":"gradient-stationary-audit","semantic_template":"qf-mod44-gradient-stationary-audit/{\"dimension\":2,\"point\":[[0,1],[0,1]],\"terms\":[[[25,1],[2,0]],[[26,1],[0,2]]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r06-gradient-vector-0076","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-boundary-counterexample；資料={\"mode\":\"incomplete_coordinate_list\",\"scale\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(incomplete_coordinate_list, 3, [1*partial_x, 1*partial_y], 2, 否, missing_coordinate_component_is_not_a_gradient_in_R3)","explanation_zh":"Wave6第七層核對=(incomplete_coordinate_list, 3, [1*partial_x, 1*partial_y], 2, 否, missing_coordinate_component_is_not_a_gradient_in_R3)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod44-gradient-boundary-counterexample","seed":202611136076,"parameters":{"mode":"incomplete_coordinate_list","scale":1,"domain":"Exact_polynomial_gradient_vector","operation":"gradient-boundary-counterexample","semantic_template":"qf-mod44-gradient-boundary-counterexample/{\"mode\":\"incomplete_coordinate_list\",\"scale\":[1,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r06-gradient-vector-0077","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-boundary-counterexample；資料={\"mode\":\"nondifferentiable_norm\",\"scale\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(nondifferentiable_norm, f(x,y)=2*sqrt(x^2+y^2), [0, 0], 否, 否, coordinate_partials_fail_at_the_origin_so_no_gradient_is_defined)","explanation_zh":"Wave6第七層核對=(nondifferentiable_norm, f(x,y)=2*sqrt(x^2+y^2), [0, 0], 否, 否, coordinate_partials_fail_at_the_origin_so_no_gradient_is_defined)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod44-gradient-boundary-counterexample","seed":202611136077,"parameters":{"mode":"nondifferentiable_norm","scale":2,"domain":"Exact_polynomial_gradient_vector","operation":"gradient-boundary-counterexample","semantic_template":"qf-mod44-gradient-boundary-counterexample/{\"mode\":\"nondifferentiable_norm\",\"scale\":[2,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r06-gradient-vector-0078","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-boundary-counterexample；資料={\"mode\":\"partials_exist_not_differentiable\",\"scale\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(partials_exist_not_differentiable, f(x,y)=3*x^3/(x^2+y^2)_off_origin;f(0,0)=0, [0, 0], [3, 0], 是, along_y=x:f(x,x)=3/2*x_but_linear_candidate=3*x, 否, both_coordinate_partials_exist_but_the_diagonal_remainder_is_not_little_o_of_the_norm)","explanation_zh":"Wave6第七層核對=(partials_exist_not_differentiable, f(x,y)=3*x^3/(x^2+y^2)_off_origin;f(0,0)=0, [0, 0], [3, 0], 是, along_y=x:f(x,x)=3/2*x_but_linear_candidate=3*x, 否, both_coordinate_partials_exist_but_the_diagonal_remainder_is_not_little_o_of_the_norm)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod44-gradient-boundary-counterexample","seed":202611136078,"parameters":{"mode":"partials_exist_not_differentiable","scale":3,"domain":"Exact_polynomial_gradient_vector","operation":"gradient-boundary-counterexample","semantic_template":"qf-mod44-gradient-boundary-counterexample/{\"mode\":\"partials_exist_not_differentiable\",\"scale\":[3,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r06-gradient-vector-0079","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-boundary-counterexample；資料={\"mode\":\"saddle_stationary\",\"scale\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(saddle_stationary, f(x,y)=4*(x^2-y^2), [0, 0], [0, 0], 是, 否, zero_gradient_at_a_saddle_is_not_an_extremum_certificate)","explanation_zh":"Wave6第七層核對=(saddle_stationary, f(x,y)=4*(x^2-y^2), [0, 0], [0, 0], 是, 否, zero_gradient_at_a_saddle_is_not_an_extremum_certificate)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod44-gradient-boundary-counterexample","seed":202611136079,"parameters":{"mode":"saddle_stationary","scale":4,"domain":"Exact_polynomial_gradient_vector","operation":"gradient-boundary-counterexample","semantic_template":"qf-mod44-gradient-boundary-counterexample/{\"mode\":\"saddle_stationary\",\"scale\":[4,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r06-gradient-vector-0080","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-boundary-counterexample；資料={\"mode\":\"incomplete_coordinate_list\",\"scale\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(incomplete_coordinate_list, 3, [5*partial_x, 5*partial_y], 2, 否, missing_coordinate_component_is_not_a_gradient_in_R3)","explanation_zh":"Wave6第七層核對=(incomplete_coordinate_list, 3, [5*partial_x, 5*partial_y], 2, 否, missing_coordinate_component_is_not_a_gradient_in_R3)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod44-gradient-boundary-counterexample","seed":202611136080,"parameters":{"mode":"incomplete_coordinate_list","scale":5,"domain":"Exact_polynomial_gradient_vector","operation":"gradient-boundary-counterexample","semantic_template":"qf-mod44-gradient-boundary-counterexample/{\"mode\":\"incomplete_coordinate_list\",\"scale\":[5,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r06-gradient-vector-0081","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-boundary-counterexample；資料={\"mode\":\"nondifferentiable_norm\",\"scale\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(nondifferentiable_norm, f(x,y)=6*sqrt(x^2+y^2), [0, 0], 否, 否, coordinate_partials_fail_at_the_origin_so_no_gradient_is_defined)","explanation_zh":"Wave6第七層核對=(nondifferentiable_norm, f(x,y)=6*sqrt(x^2+y^2), [0, 0], 否, 否, coordinate_partials_fail_at_the_origin_so_no_gradient_is_defined)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod44-gradient-boundary-counterexample","seed":202611136081,"parameters":{"mode":"nondifferentiable_norm","scale":6,"domain":"Exact_polynomial_gradient_vector","operation":"gradient-boundary-counterexample","semantic_template":"qf-mod44-gradient-boundary-counterexample/{\"mode\":\"nondifferentiable_norm\",\"scale\":[6,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r06-gradient-vector-0082","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-boundary-counterexample；資料={\"mode\":\"partials_exist_not_differentiable\",\"scale\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(partials_exist_not_differentiable, f(x,y)=7*x^3/(x^2+y^2)_off_origin;f(0,0)=0, [0, 0], [7, 0], 是, along_y=x:f(x,x)=7/2*x_but_linear_candidate=7*x, 否, both_coordinate_partials_exist_but_the_diagonal_remainder_is_not_little_o_of_the_norm)","explanation_zh":"Wave6第七層核對=(partials_exist_not_differentiable, f(x,y)=7*x^3/(x^2+y^2)_off_origin;f(0,0)=0, [0, 0], [7, 0], 是, along_y=x:f(x,x)=7/2*x_but_linear_candidate=7*x, 否, both_coordinate_partials_exist_but_the_diagonal_remainder_is_not_little_o_of_the_norm)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod44-gradient-boundary-counterexample","seed":202611136082,"parameters":{"mode":"partials_exist_not_differentiable","scale":7,"domain":"Exact_polynomial_gradient_vector","operation":"gradient-boundary-counterexample","semantic_template":"qf-mod44-gradient-boundary-counterexample/{\"mode\":\"partials_exist_not_differentiable\",\"scale\":[7,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r06-gradient-vector-0083","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-boundary-counterexample；資料={\"mode\":\"saddle_stationary\",\"scale\":8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(saddle_stationary, f(x,y)=8*(x^2-y^2), [0, 0], [0, 0], 是, 否, zero_gradient_at_a_saddle_is_not_an_extremum_certificate)","explanation_zh":"Wave6第七層核對=(saddle_stationary, f(x,y)=8*(x^2-y^2), [0, 0], [0, 0], 是, 否, zero_gradient_at_a_saddle_is_not_an_extremum_certificate)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod44-gradient-boundary-counterexample","seed":202611136083,"parameters":{"mode":"saddle_stationary","scale":8,"domain":"Exact_polynomial_gradient_vector","operation":"gradient-boundary-counterexample","semantic_template":"qf-mod44-gradient-boundary-counterexample/{\"mode\":\"saddle_stationary\",\"scale\":[8,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r06-gradient-vector-0084","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-boundary-counterexample；資料={\"mode\":\"incomplete_coordinate_list\",\"scale\":9}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(incomplete_coordinate_list, 3, [9*partial_x, 9*partial_y], 2, 否, missing_coordinate_component_is_not_a_gradient_in_R3)","explanation_zh":"Wave6第七層核對=(incomplete_coordinate_list, 3, [9*partial_x, 9*partial_y], 2, 否, missing_coordinate_component_is_not_a_gradient_in_R3)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod44-gradient-boundary-counterexample","seed":202611136084,"parameters":{"mode":"incomplete_coordinate_list","scale":9,"domain":"Exact_polynomial_gradient_vector","operation":"gradient-boundary-counterexample","semantic_template":"qf-mod44-gradient-boundary-counterexample/{\"mode\":\"incomplete_coordinate_list\",\"scale\":[9,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r06-gradient-vector-0085","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-boundary-counterexample；資料={\"mode\":\"nondifferentiable_norm\",\"scale\":10}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(nondifferentiable_norm, f(x,y)=10*sqrt(x^2+y^2), [0, 0], 否, 否, coordinate_partials_fail_at_the_origin_so_no_gradient_is_defined)","explanation_zh":"Wave6第七層核對=(nondifferentiable_norm, f(x,y)=10*sqrt(x^2+y^2), [0, 0], 否, 否, coordinate_partials_fail_at_the_origin_so_no_gradient_is_defined)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod44-gradient-boundary-counterexample","seed":202611136085,"parameters":{"mode":"nondifferentiable_norm","scale":10,"domain":"Exact_polynomial_gradient_vector","operation":"gradient-boundary-counterexample","semantic_template":"qf-mod44-gradient-boundary-counterexample/{\"mode\":\"nondifferentiable_norm\",\"scale\":[10,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r06-gradient-vector-0086","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-boundary-counterexample；資料={\"mode\":\"partials_exist_not_differentiable\",\"scale\":11}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(partials_exist_not_differentiable, f(x,y)=11*x^3/(x^2+y^2)_off_origin;f(0,0)=0, [0, 0], [11, 0], 是, along_y=x:f(x,x)=11/2*x_but_linear_candidate=11*x, 否, both_coordinate_partials_exist_but_the_diagonal_remainder_is_not_little_o_of_the_norm)","explanation_zh":"Wave6第七層核對=(partials_exist_not_differentiable, f(x,y)=11*x^3/(x^2+y^2)_off_origin;f(0,0)=0, [0, 0], [11, 0], 是, along_y=x:f(x,x)=11/2*x_but_linear_candidate=11*x, 否, both_coordinate_partials_exist_but_the_diagonal_remainder_is_not_little_o_of_the_norm)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod44-gradient-boundary-counterexample","seed":202611136086,"parameters":{"mode":"partials_exist_not_differentiable","scale":11,"domain":"Exact_polynomial_gradient_vector","operation":"gradient-boundary-counterexample","semantic_template":"qf-mod44-gradient-boundary-counterexample/{\"mode\":\"partials_exist_not_differentiable\",\"scale\":[11,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r06-gradient-vector-0087","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-boundary-counterexample；資料={\"mode\":\"saddle_stationary\",\"scale\":12}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(saddle_stationary, f(x,y)=12*(x^2-y^2), [0, 0], [0, 0], 是, 否, zero_gradient_at_a_saddle_is_not_an_extremum_certificate)","explanation_zh":"Wave6第七層核對=(saddle_stationary, f(x,y)=12*(x^2-y^2), [0, 0], [0, 0], 是, 否, zero_gradient_at_a_saddle_is_not_an_extremum_certificate)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod44-gradient-boundary-counterexample","seed":202611136087,"parameters":{"mode":"saddle_stationary","scale":12,"domain":"Exact_polynomial_gradient_vector","operation":"gradient-boundary-counterexample","semantic_template":"qf-mod44-gradient-boundary-counterexample/{\"mode\":\"saddle_stationary\",\"scale\":[12,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r06-gradient-vector-0088","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-boundary-counterexample；資料={\"mode\":\"incomplete_coordinate_list\",\"scale\":13}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(incomplete_coordinate_list, 3, [13*partial_x, 13*partial_y], 2, 否, missing_coordinate_component_is_not_a_gradient_in_R3)","explanation_zh":"Wave6第七層核對=(incomplete_coordinate_list, 3, [13*partial_x, 13*partial_y], 2, 否, missing_coordinate_component_is_not_a_gradient_in_R3)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod44-gradient-boundary-counterexample","seed":202611136088,"parameters":{"mode":"incomplete_coordinate_list","scale":13,"domain":"Exact_polynomial_gradient_vector","operation":"gradient-boundary-counterexample","semantic_template":"qf-mod44-gradient-boundary-counterexample/{\"mode\":\"incomplete_coordinate_list\",\"scale\":[13,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r06-gradient-vector-0089","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-boundary-counterexample；資料={\"mode\":\"nondifferentiable_norm\",\"scale\":14}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(nondifferentiable_norm, f(x,y)=14*sqrt(x^2+y^2), [0, 0], 否, 否, coordinate_partials_fail_at_the_origin_so_no_gradient_is_defined)","explanation_zh":"Wave6第七層核對=(nondifferentiable_norm, f(x,y)=14*sqrt(x^2+y^2), [0, 0], 否, 否, coordinate_partials_fail_at_the_origin_so_no_gradient_is_defined)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod44-gradient-boundary-counterexample","seed":202611136089,"parameters":{"mode":"nondifferentiable_norm","scale":14,"domain":"Exact_polynomial_gradient_vector","operation":"gradient-boundary-counterexample","semantic_template":"qf-mod44-gradient-boundary-counterexample/{\"mode\":\"nondifferentiable_norm\",\"scale\":[14,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r06-gradient-vector-0090","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-boundary-counterexample；資料={\"mode\":\"partials_exist_not_differentiable\",\"scale\":15}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(partials_exist_not_differentiable, f(x,y)=15*x^3/(x^2+y^2)_off_origin;f(0,0)=0, [0, 0], [15, 0], 是, along_y=x:f(x,x)=15/2*x_but_linear_candidate=15*x, 否, both_coordinate_partials_exist_but_the_diagonal_remainder_is_not_little_o_of_the_norm)","explanation_zh":"Wave6第七層核對=(partials_exist_not_differentiable, f(x,y)=15*x^3/(x^2+y^2)_off_origin;f(0,0)=0, [0, 0], [15, 0], 是, along_y=x:f(x,x)=15/2*x_but_linear_candidate=15*x, 否, both_coordinate_partials_exist_but_the_diagonal_remainder_is_not_little_o_of_the_norm)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod44-gradient-boundary-counterexample","seed":202611136090,"parameters":{"mode":"partials_exist_not_differentiable","scale":15,"domain":"Exact_polynomial_gradient_vector","operation":"gradient-boundary-counterexample","semantic_template":"qf-mod44-gradient-boundary-counterexample/{\"mode\":\"partials_exist_not_differentiable\",\"scale\":[15,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r06-gradient-vector-0091","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-boundary-counterexample；資料={\"mode\":\"saddle_stationary\",\"scale\":16}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(saddle_stationary, f(x,y)=16*(x^2-y^2), [0, 0], [0, 0], 是, 否, zero_gradient_at_a_saddle_is_not_an_extremum_certificate)","explanation_zh":"Wave6第七層核對=(saddle_stationary, f(x,y)=16*(x^2-y^2), [0, 0], [0, 0], 是, 否, zero_gradient_at_a_saddle_is_not_an_extremum_certificate)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod44-gradient-boundary-counterexample","seed":202611136091,"parameters":{"mode":"saddle_stationary","scale":16,"domain":"Exact_polynomial_gradient_vector","operation":"gradient-boundary-counterexample","semantic_template":"qf-mod44-gradient-boundary-counterexample/{\"mode\":\"saddle_stationary\",\"scale\":[16,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r06-gradient-vector-0092","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-boundary-counterexample；資料={\"mode\":\"incomplete_coordinate_list\",\"scale\":17}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(incomplete_coordinate_list, 3, [17*partial_x, 17*partial_y], 2, 否, missing_coordinate_component_is_not_a_gradient_in_R3)","explanation_zh":"Wave6第七層核對=(incomplete_coordinate_list, 3, [17*partial_x, 17*partial_y], 2, 否, missing_coordinate_component_is_not_a_gradient_in_R3)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod44-gradient-boundary-counterexample","seed":202611136092,"parameters":{"mode":"incomplete_coordinate_list","scale":17,"domain":"Exact_polynomial_gradient_vector","operation":"gradient-boundary-counterexample","semantic_template":"qf-mod44-gradient-boundary-counterexample/{\"mode\":\"incomplete_coordinate_list\",\"scale\":[17,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r06-gradient-vector-0093","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-boundary-counterexample；資料={\"mode\":\"nondifferentiable_norm\",\"scale\":18}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(nondifferentiable_norm, f(x,y)=18*sqrt(x^2+y^2), [0, 0], 否, 否, coordinate_partials_fail_at_the_origin_so_no_gradient_is_defined)","explanation_zh":"Wave6第七層核對=(nondifferentiable_norm, f(x,y)=18*sqrt(x^2+y^2), [0, 0], 否, 否, coordinate_partials_fail_at_the_origin_so_no_gradient_is_defined)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod44-gradient-boundary-counterexample","seed":202611136093,"parameters":{"mode":"nondifferentiable_norm","scale":18,"domain":"Exact_polynomial_gradient_vector","operation":"gradient-boundary-counterexample","semantic_template":"qf-mod44-gradient-boundary-counterexample/{\"mode\":\"nondifferentiable_norm\",\"scale\":[18,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r06-gradient-vector-0094","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-boundary-counterexample；資料={\"mode\":\"partials_exist_not_differentiable\",\"scale\":19}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(partials_exist_not_differentiable, f(x,y)=19*x^3/(x^2+y^2)_off_origin;f(0,0)=0, [0, 0], [19, 0], 是, along_y=x:f(x,x)=19/2*x_but_linear_candidate=19*x, 否, both_coordinate_partials_exist_but_the_diagonal_remainder_is_not_little_o_of_the_norm)","explanation_zh":"Wave6第七層核對=(partials_exist_not_differentiable, f(x,y)=19*x^3/(x^2+y^2)_off_origin;f(0,0)=0, [0, 0], [19, 0], 是, along_y=x:f(x,x)=19/2*x_but_linear_candidate=19*x, 否, both_coordinate_partials_exist_but_the_diagonal_remainder_is_not_little_o_of_the_norm)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod44-gradient-boundary-counterexample","seed":202611136094,"parameters":{"mode":"partials_exist_not_differentiable","scale":19,"domain":"Exact_polynomial_gradient_vector","operation":"gradient-boundary-counterexample","semantic_template":"qf-mod44-gradient-boundary-counterexample/{\"mode\":\"partials_exist_not_differentiable\",\"scale\":[19,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r06-gradient-vector-0095","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-boundary-counterexample；資料={\"mode\":\"saddle_stationary\",\"scale\":20}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(saddle_stationary, f(x,y)=20*(x^2-y^2), [0, 0], [0, 0], 是, 否, zero_gradient_at_a_saddle_is_not_an_extremum_certificate)","explanation_zh":"Wave6第七層核對=(saddle_stationary, f(x,y)=20*(x^2-y^2), [0, 0], [0, 0], 是, 否, zero_gradient_at_a_saddle_is_not_an_extremum_certificate)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod44-gradient-boundary-counterexample","seed":202611136095,"parameters":{"mode":"saddle_stationary","scale":20,"domain":"Exact_polynomial_gradient_vector","operation":"gradient-boundary-counterexample","semantic_template":"qf-mod44-gradient-boundary-counterexample/{\"mode\":\"saddle_stationary\",\"scale\":[20,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r06-gradient-vector-0096","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-boundary-counterexample；資料={\"mode\":\"incomplete_coordinate_list\",\"scale\":21}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(incomplete_coordinate_list, 3, [21*partial_x, 21*partial_y], 2, 否, missing_coordinate_component_is_not_a_gradient_in_R3)","explanation_zh":"Wave6第七層核對=(incomplete_coordinate_list, 3, [21*partial_x, 21*partial_y], 2, 否, missing_coordinate_component_is_not_a_gradient_in_R3)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod44-gradient-boundary-counterexample","seed":202611136096,"parameters":{"mode":"incomplete_coordinate_list","scale":21,"domain":"Exact_polynomial_gradient_vector","operation":"gradient-boundary-counterexample","semantic_template":"qf-mod44-gradient-boundary-counterexample/{\"mode\":\"incomplete_coordinate_list\",\"scale\":[21,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r06-gradient-vector-0097","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-boundary-counterexample；資料={\"mode\":\"nondifferentiable_norm\",\"scale\":22}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(nondifferentiable_norm, f(x,y)=22*sqrt(x^2+y^2), [0, 0], 否, 否, coordinate_partials_fail_at_the_origin_so_no_gradient_is_defined)","explanation_zh":"Wave6第七層核對=(nondifferentiable_norm, f(x,y)=22*sqrt(x^2+y^2), [0, 0], 否, 否, coordinate_partials_fail_at_the_origin_so_no_gradient_is_defined)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod44-gradient-boundary-counterexample","seed":202611136097,"parameters":{"mode":"nondifferentiable_norm","scale":22,"domain":"Exact_polynomial_gradient_vector","operation":"gradient-boundary-counterexample","semantic_template":"qf-mod44-gradient-boundary-counterexample/{\"mode\":\"nondifferentiable_norm\",\"scale\":[22,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r06-gradient-vector-0098","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-boundary-counterexample；資料={\"mode\":\"partials_exist_not_differentiable\",\"scale\":23}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(partials_exist_not_differentiable, f(x,y)=23*x^3/(x^2+y^2)_off_origin;f(0,0)=0, [0, 0], [23, 0], 是, along_y=x:f(x,x)=23/2*x_but_linear_candidate=23*x, 否, both_coordinate_partials_exist_but_the_diagonal_remainder_is_not_little_o_of_the_norm)","explanation_zh":"Wave6第七層核對=(partials_exist_not_differentiable, f(x,y)=23*x^3/(x^2+y^2)_off_origin;f(0,0)=0, [0, 0], [23, 0], 是, along_y=x:f(x,x)=23/2*x_but_linear_candidate=23*x, 否, both_coordinate_partials_exist_but_the_diagonal_remainder_is_not_little_o_of_the_norm)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod44-gradient-boundary-counterexample","seed":202611136098,"parameters":{"mode":"partials_exist_not_differentiable","scale":23,"domain":"Exact_polynomial_gradient_vector","operation":"gradient-boundary-counterexample","semantic_template":"qf-mod44-gradient-boundary-counterexample/{\"mode\":\"partials_exist_not_differentiable\",\"scale\":[23,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r06-gradient-vector-0099","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-boundary-counterexample；資料={\"mode\":\"saddle_stationary\",\"scale\":24}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(saddle_stationary, f(x,y)=24*(x^2-y^2), [0, 0], [0, 0], 是, 否, zero_gradient_at_a_saddle_is_not_an_extremum_certificate)","explanation_zh":"Wave6第七層核對=(saddle_stationary, f(x,y)=24*(x^2-y^2), [0, 0], [0, 0], 是, 否, zero_gradient_at_a_saddle_is_not_an_extremum_certificate)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod44-gradient-boundary-counterexample","seed":202611136099,"parameters":{"mode":"saddle_stationary","scale":24,"domain":"Exact_polynomial_gradient_vector","operation":"gradient-boundary-counterexample","semantic_template":"qf-mod44-gradient-boundary-counterexample/{\"mode\":\"saddle_stationary\",\"scale\":[24,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261113-r06-gradient-vector-0100","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","language":"zh-Hant","source_mko_ids":["mko-gradient-vector"],"title_zh":"梯度向量","stem_zh":"standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-boundary-counterexample；資料={\"mode\":\"incomplete_coordinate_list\",\"scale\":25}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(incomplete_coordinate_list, 3, [25*partial_x, 25*partial_y], 2, 否, missing_coordinate_component_is_not_a_gradient_in_R3)","explanation_zh":"Wave6第七層核對=(incomplete_coordinate_list, 3, [25*partial_x, 25*partial_y], 2, 否, missing_coordinate_component_is_not_a_gradient_in_R3)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["梯度向量"]},"variant":{"family_id":"qf-mod44-gradient-boundary-counterexample","seed":202611136100,"parameters":{"mode":"incomplete_coordinate_list","scale":25,"domain":"Exact_polynomial_gradient_vector","operation":"gradient-boundary-counterexample","semantic_template":"qf-mod44-gradient-boundary-counterexample/{\"mode\":\"incomplete_coordinate_list\",\"scale\":[25,1]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}