{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r04-0001","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-multiplier-condition"],"title_zh":"Lagrange乘子條件","stem_zh":"feasibility、constraint-gradient full row rank與grad_f=J_g^T*lambda convention。操作=lagrange-regularity-audit；明示資料={\"constraint_gradients\":[[1,0],[0,1]]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], 2, 2, 2, 是, full_row_rank_means_constraint_gradients_are_linearly_independent)","explanation_zh":"Wave6第十三層核對=([[1, 0], [0, 1]], 2, 2, 2, 是, full_row_rank_means_constraint_gradients_are_linearly_independent)；依完整明示資料精確重算；regular equality-constrained local extremum只得到必要multiplier condition；stationarity非充分；rank deficient是不適用而非無極值，且本條件不是inequality KKT、global或uniqueness證書。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Lagrange乘子條件"]},"variant":{"family_id":"qf-mod50-lagrange-regularity-audit","seed":202611194001,"parameters":{"constraint_gradients":[[1,0],[0,1]],"domain":"Exact_equality_constraint_Lagrange_condition","operation":"lagrange-regularity-audit","semantic_template":"lagrange-regularity-audit|{\"constraint_gradients\":[[[0,1],[1,1]],[[1,1],[0,1]]]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r04-0002","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-multiplier-condition"],"title_zh":"Lagrange乘子條件","stem_zh":"feasibility、constraint-gradient full row rank與grad_f=J_g^T*lambda convention。操作=lagrange-regularity-audit；明示資料={\"constraint_gradients\":[[1,0],[0,2]]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 2]], 2, 2, 2, 是, full_row_rank_means_constraint_gradients_are_linearly_independent)","explanation_zh":"Wave6第十三層核對=([[1, 0], [0, 2]], 2, 2, 2, 是, full_row_rank_means_constraint_gradients_are_linearly_independent)；依完整明示資料精確重算；regular equality-constrained local extremum只得到必要multiplier condition；stationarity非充分；rank deficient是不適用而非無極值，且本條件不是inequality KKT、global或uniqueness證書。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Lagrange乘子條件"]},"variant":{"family_id":"qf-mod50-lagrange-regularity-audit","seed":202611194002,"parameters":{"constraint_gradients":[[1,0],[0,2]],"domain":"Exact_equality_constraint_Lagrange_condition","operation":"lagrange-regularity-audit","semantic_template":"lagrange-regularity-audit|{\"constraint_gradients\":[[[0,1],[2,1]],[[1,1],[0,1]]]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r04-0003","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-multiplier-condition"],"title_zh":"Lagrange乘子條件","stem_zh":"feasibility、constraint-gradient full row rank與grad_f=J_g^T*lambda convention。操作=lagrange-regularity-audit；明示資料={\"constraint_gradients\":[[1,0],[0,3]]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 3]], 2, 2, 2, 是, full_row_rank_means_constraint_gradients_are_linearly_independent)","explanation_zh":"Wave6第十三層核對=([[1, 0], [0, 3]], 2, 2, 2, 是, full_row_rank_means_constraint_gradients_are_linearly_independent)；依完整明示資料精確重算；regular equality-constrained local extremum只得到必要multiplier condition；stationarity非充分；rank deficient是不適用而非無極值，且本條件不是inequality KKT、global或uniqueness證書。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Lagrange乘子條件"]},"variant":{"family_id":"qf-mod50-lagrange-regularity-audit","seed":202611194003,"parameters":{"constraint_gradients":[[1,0],[0,3]],"domain":"Exact_equality_constraint_Lagrange_condition","operation":"lagrange-regularity-audit","semantic_template":"lagrange-regularity-audit|{\"constraint_gradients\":[[[0,1],[3,1]],[[1,1],[0,1]]]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r04-0004","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-multiplier-condition"],"title_zh":"Lagrange乘子條件","stem_zh":"feasibility、constraint-gradient full row rank與grad_f=J_g^T*lambda convention。操作=lagrange-regularity-audit；明示資料={\"constraint_gradients\":[[1,0],[0,4]]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 4]], 2, 2, 2, 是, full_row_rank_means_constraint_gradients_are_linearly_independent)","explanation_zh":"Wave6第十三層核對=([[1, 0], [0, 4]], 2, 2, 2, 是, full_row_rank_means_constraint_gradients_are_linearly_independent)；依完整明示資料精確重算；regular equality-constrained local extremum只得到必要multiplier condition；stationarity非充分；rank deficient是不適用而非無極值，且本條件不是inequality KKT、global或uniqueness證書。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Lagrange乘子條件"]},"variant":{"family_id":"qf-mod50-lagrange-regularity-audit","seed":202611194004,"parameters":{"constraint_gradients":[[1,0],[0,4]],"domain":"Exact_equality_constraint_Lagrange_condition","operation":"lagrange-regularity-audit","semantic_template":"lagrange-regularity-audit|{\"constraint_gradients\":[[[0,1],[4,1]],[[1,1],[0,1]]]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r04-0005","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-multiplier-condition"],"title_zh":"Lagrange乘子條件","stem_zh":"feasibility、constraint-gradient full row rank與grad_f=J_g^T*lambda convention。操作=lagrange-regularity-audit；明示資料={\"constraint_gradients\":[[1,0],[0,5]]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 5]], 2, 2, 2, 是, full_row_rank_means_constraint_gradients_are_linearly_independent)","explanation_zh":"Wave6第十三層核對=([[1, 0], [0, 5]], 2, 2, 2, 是, full_row_rank_means_constraint_gradients_are_linearly_independent)；依完整明示資料精確重算；regular equality-constrained local extremum只得到必要multiplier condition；stationarity非充分；rank deficient是不適用而非無極值，且本條件不是inequality KKT、global或uniqueness證書。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Lagrange乘子條件"]},"variant":{"family_id":"qf-mod50-lagrange-regularity-audit","seed":202611194005,"parameters":{"constraint_gradients":[[1,0],[0,5]],"domain":"Exact_equality_constraint_Lagrange_condition","operation":"lagrange-regularity-audit","semantic_template":"lagrange-regularity-audit|{\"constraint_gradients\":[[[0,1],[5,1]],[[1,1],[0,1]]]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r04-0006","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-multiplier-condition"],"title_zh":"Lagrange乘子條件","stem_zh":"feasibility、constraint-gradient full row rank與grad_f=J_g^T*lambda convention。操作=lagrange-regularity-audit；明示資料={\"constraint_gradients\":[[1,0],[0,6]]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 6]], 2, 2, 2, 是, full_row_rank_means_constraint_gradients_are_linearly_independent)","explanation_zh":"Wave6第十三層核對=([[1, 0], [0, 6]], 2, 2, 2, 是, full_row_rank_means_constraint_gradients_are_linearly_independent)；依完整明示資料精確重算；regular equality-constrained local extremum只得到必要multiplier condition；stationarity非充分；rank deficient是不適用而非無極值，且本條件不是inequality KKT、global或uniqueness證書。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Lagrange乘子條件"]},"variant":{"family_id":"qf-mod50-lagrange-regularity-audit","seed":202611194006,"parameters":{"constraint_gradients":[[1,0],[0,6]],"domain":"Exact_equality_constraint_Lagrange_condition","operation":"lagrange-regularity-audit","semantic_template":"lagrange-regularity-audit|{\"constraint_gradients\":[[[0,1],[6,1]],[[1,1],[0,1]]]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r04-0007","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-multiplier-condition"],"title_zh":"Lagrange乘子條件","stem_zh":"feasibility、constraint-gradient full row rank與grad_f=J_g^T*lambda convention。操作=lagrange-regularity-audit；明示資料={\"constraint_gradients\":[[1,0],[0,7]]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 7]], 2, 2, 2, 是, full_row_rank_means_constraint_gradients_are_linearly_independent)","explanation_zh":"Wave6第十三層核對=([[1, 0], [0, 7]], 2, 2, 2, 是, full_row_rank_means_constraint_gradients_are_linearly_independent)；依完整明示資料精確重算；regular equality-constrained local extremum只得到必要multiplier condition；stationarity非充分；rank deficient是不適用而非無極值，且本條件不是inequality KKT、global或uniqueness證書。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Lagrange乘子條件"]},"variant":{"family_id":"qf-mod50-lagrange-regularity-audit","seed":202611194007,"parameters":{"constraint_gradients":[[1,0],[0,7]],"domain":"Exact_equality_constraint_Lagrange_condition","operation":"lagrange-regularity-audit","semantic_template":"lagrange-regularity-audit|{\"constraint_gradients\":[[[0,1],[7,1]],[[1,1],[0,1]]]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r04-0008","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-multiplier-condition"],"title_zh":"Lagrange乘子條件","stem_zh":"feasibility、constraint-gradient full row rank與grad_f=J_g^T*lambda convention。操作=lagrange-regularity-audit；明示資料={\"constraint_gradients\":[[1,0],[0,8]]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 8]], 2, 2, 2, 是, full_row_rank_means_constraint_gradients_are_linearly_independent)","explanation_zh":"Wave6第十三層核對=([[1, 0], [0, 8]], 2, 2, 2, 是, full_row_rank_means_constraint_gradients_are_linearly_independent)；依完整明示資料精確重算；regular equality-constrained local extremum只得到必要multiplier condition；stationarity非充分；rank deficient是不適用而非無極值，且本條件不是inequality KKT、global或uniqueness證書。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Lagrange乘子條件"]},"variant":{"family_id":"qf-mod50-lagrange-regularity-audit","seed":202611194008,"parameters":{"constraint_gradients":[[1,0],[0,8]],"domain":"Exact_equality_constraint_Lagrange_condition","operation":"lagrange-regularity-audit","semantic_template":"lagrange-regularity-audit|{\"constraint_gradients\":[[[0,1],[8,1]],[[1,1],[0,1]]]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r04-0009","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-multiplier-condition"],"title_zh":"Lagrange乘子條件","stem_zh":"feasibility、constraint-gradient full row rank與grad_f=J_g^T*lambda convention。操作=lagrange-regularity-audit；明示資料={\"constraint_gradients\":[[1,0],[0,9]]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 9]], 2, 2, 2, 是, full_row_rank_means_constraint_gradients_are_linearly_independent)","explanation_zh":"Wave6第十三層核對=([[1, 0], [0, 9]], 2, 2, 2, 是, full_row_rank_means_constraint_gradients_are_linearly_independent)；依完整明示資料精確重算；regular equality-constrained local extremum只得到必要multiplier condition；stationarity非充分；rank deficient是不適用而非無極值，且本條件不是inequality KKT、global或uniqueness證書。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Lagrange乘子條件"]},"variant":{"family_id":"qf-mod50-lagrange-regularity-audit","seed":202611194009,"parameters":{"constraint_gradients":[[1,0],[0,9]],"domain":"Exact_equality_constraint_Lagrange_condition","operation":"lagrange-regularity-audit","semantic_template":"lagrange-regularity-audit|{\"constraint_gradients\":[[[0,1],[9,1]],[[1,1],[0,1]]]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r04-0010","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-multiplier-condition"],"title_zh":"Lagrange乘子條件","stem_zh":"feasibility、constraint-gradient full row rank與grad_f=J_g^T*lambda convention。操作=lagrange-regularity-audit；明示資料={\"constraint_gradients\":[[1,0],[0,10]]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 10]], 2, 2, 2, 是, full_row_rank_means_constraint_gradients_are_linearly_independent)","explanation_zh":"Wave6第十三層核對=([[1, 0], [0, 10]], 2, 2, 2, 是, full_row_rank_means_constraint_gradients_are_linearly_independent)；依完整明示資料精確重算；regular equality-constrained local extremum只得到必要multiplier condition；stationarity非充分；rank deficient是不適用而非無極值，且本條件不是inequality KKT、global或uniqueness證書。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Lagrange乘子條件"]},"variant":{"family_id":"qf-mod50-lagrange-regularity-audit","seed":202611194010,"parameters":{"constraint_gradients":[[1,0],[0,10]],"domain":"Exact_equality_constraint_Lagrange_condition","operation":"lagrange-regularity-audit","semantic_template":"lagrange-regularity-audit|{\"constraint_gradients\":[[[0,1],[10,1]],[[1,1],[0,1]]]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r04-0011","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-multiplier-condition"],"title_zh":"Lagrange乘子條件","stem_zh":"feasibility、constraint-gradient full row rank與grad_f=J_g^T*lambda convention。操作=lagrange-regularity-audit；明示資料={\"constraint_gradients\":[[1,0],[0,11]]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 11]], 2, 2, 2, 是, full_row_rank_means_constraint_gradients_are_linearly_independent)","explanation_zh":"Wave6第十三層核對=([[1, 0], [0, 11]], 2, 2, 2, 是, full_row_rank_means_constraint_gradients_are_linearly_independent)；依完整明示資料精確重算；regular equality-constrained local extremum只得到必要multiplier condition；stationarity非充分；rank deficient是不適用而非無極值，且本條件不是inequality KKT、global或uniqueness證書。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Lagrange乘子條件"]},"variant":{"family_id":"qf-mod50-lagrange-regularity-audit","seed":202611194011,"parameters":{"constraint_gradients":[[1,0],[0,11]],"domain":"Exact_equality_constraint_Lagrange_condition","operation":"lagrange-regularity-audit","semantic_template":"lagrange-regularity-audit|{\"constraint_gradients\":[[[0,1],[11,1]],[[1,1],[0,1]]]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r04-0012","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-multiplier-condition"],"title_zh":"Lagrange乘子條件","stem_zh":"feasibility、constraint-gradient full row rank與grad_f=J_g^T*lambda convention。操作=lagrange-regularity-audit；明示資料={\"constraint_gradients\":[[1,0],[0,12]]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 12]], 2, 2, 2, 是, full_row_rank_means_constraint_gradients_are_linearly_independent)","explanation_zh":"Wave6第十三層核對=([[1, 0], [0, 12]], 2, 2, 2, 是, full_row_rank_means_constraint_gradients_are_linearly_independent)；依完整明示資料精確重算；regular equality-constrained local extremum只得到必要multiplier condition；stationarity非充分；rank deficient是不適用而非無極值，且本條件不是inequality KKT、global或uniqueness證書。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Lagrange乘子條件"]},"variant":{"family_id":"qf-mod50-lagrange-regularity-audit","seed":202611194012,"parameters":{"constraint_gradients":[[1,0],[0,12]],"domain":"Exact_equality_constraint_Lagrange_condition","operation":"lagrange-regularity-audit","semantic_template":"lagrange-regularity-audit|{\"constraint_gradients\":[[[0,1],[12,1]],[[1,1],[0,1]]]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r04-0013","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-multiplier-condition"],"title_zh":"Lagrange乘子條件","stem_zh":"feasibility、constraint-gradient full row rank與grad_f=J_g^T*lambda convention。操作=lagrange-regularity-audit；明示資料={\"constraint_gradients\":[[1,0],[0,13]]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 13]], 2, 2, 2, 是, full_row_rank_means_constraint_gradients_are_linearly_independent)","explanation_zh":"Wave6第十三層核對=([[1, 0], [0, 13]], 2, 2, 2, 是, full_row_rank_means_constraint_gradients_are_linearly_independent)；依完整明示資料精確重算；regular equality-constrained local extremum只得到必要multiplier condition；stationarity非充分；rank deficient是不適用而非無極值，且本條件不是inequality KKT、global或uniqueness證書。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Lagrange乘子條件"]},"variant":{"family_id":"qf-mod50-lagrange-regularity-audit","seed":202611194013,"parameters":{"constraint_gradients":[[1,0],[0,13]],"domain":"Exact_equality_constraint_Lagrange_condition","operation":"lagrange-regularity-audit","semantic_template":"lagrange-regularity-audit|{\"constraint_gradients\":[[[0,1],[13,1]],[[1,1],[0,1]]]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r04-0014","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-multiplier-condition"],"title_zh":"Lagrange乘子條件","stem_zh":"feasibility、constraint-gradient full row rank與grad_f=J_g^T*lambda convention。操作=lagrange-regularity-audit；明示資料={\"constraint_gradients\":[[1,0],[0,14]]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 14]], 2, 2, 2, 是, full_row_rank_means_constraint_gradients_are_linearly_independent)","explanation_zh":"Wave6第十三層核對=([[1, 0], [0, 14]], 2, 2, 2, 是, full_row_rank_means_constraint_gradients_are_linearly_independent)；依完整明示資料精確重算；regular equality-constrained local extremum只得到必要multiplier condition；stationarity非充分；rank deficient是不適用而非無極值，且本條件不是inequality KKT、global或uniqueness證書。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Lagrange乘子條件"]},"variant":{"family_id":"qf-mod50-lagrange-regularity-audit","seed":202611194014,"parameters":{"constraint_gradients":[[1,0],[0,14]],"domain":"Exact_equality_constraint_Lagrange_condition","operation":"lagrange-regularity-audit","semantic_template":"lagrange-regularity-audit|{\"constraint_gradients\":[[[0,1],[14,1]],[[1,1],[0,1]]]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r04-0015","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-multiplier-condition"],"title_zh":"Lagrange乘子條件","stem_zh":"feasibility、constraint-gradient full row rank與grad_f=J_g^T*lambda convention。操作=lagrange-regularity-audit；明示資料={\"constraint_gradients\":[[1,0],[0,15]]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 15]], 2, 2, 2, 是, full_row_rank_means_constraint_gradients_are_linearly_independent)","explanation_zh":"Wave6第十三層核對=([[1, 0], [0, 15]], 2, 2, 2, 是, full_row_rank_means_constraint_gradients_are_linearly_independent)；依完整明示資料精確重算；regular equality-constrained local extremum只得到必要multiplier condition；stationarity非充分；rank deficient是不適用而非無極值，且本條件不是inequality KKT、global或uniqueness證書。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Lagrange乘子條件"]},"variant":{"family_id":"qf-mod50-lagrange-regularity-audit","seed":202611194015,"parameters":{"constraint_gradients":[[1,0],[0,15]],"domain":"Exact_equality_constraint_Lagrange_condition","operation":"lagrange-regularity-audit","semantic_template":"lagrange-regularity-audit|{\"constraint_gradients\":[[[0,1],[15,1]],[[1,1],[0,1]]]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r04-0016","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-multiplier-condition"],"title_zh":"Lagrange乘子條件","stem_zh":"feasibility、constraint-gradient full row rank與grad_f=J_g^T*lambda convention。操作=lagrange-regularity-audit；明示資料={\"constraint_gradients\":[[1,0],[0,16]]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 16]], 2, 2, 2, 是, full_row_rank_means_constraint_gradients_are_linearly_independent)","explanation_zh":"Wave6第十三層核對=([[1, 0], [0, 16]], 2, 2, 2, 是, full_row_rank_means_constraint_gradients_are_linearly_independent)；依完整明示資料精確重算；regular equality-constrained local extremum只得到必要multiplier condition；stationarity非充分；rank deficient是不適用而非無極值，且本條件不是inequality KKT、global或uniqueness證書。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Lagrange乘子條件"]},"variant":{"family_id":"qf-mod50-lagrange-regularity-audit","seed":202611194016,"parameters":{"constraint_gradients":[[1,0],[0,16]],"domain":"Exact_equality_constraint_Lagrange_condition","operation":"lagrange-regularity-audit","semantic_template":"lagrange-regularity-audit|{\"constraint_gradients\":[[[0,1],[16,1]],[[1,1],[0,1]]]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r04-0017","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-multiplier-condition"],"title_zh":"Lagrange乘子條件","stem_zh":"feasibility、constraint-gradient full row rank與grad_f=J_g^T*lambda convention。操作=lagrange-regularity-audit；明示資料={\"constraint_gradients\":[[1,0],[0,17]]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 17]], 2, 2, 2, 是, full_row_rank_means_constraint_gradients_are_linearly_independent)","explanation_zh":"Wave6第十三層核對=([[1, 0], [0, 17]], 2, 2, 2, 是, full_row_rank_means_constraint_gradients_are_linearly_independent)；依完整明示資料精確重算；regular equality-constrained local extremum只得到必要multiplier condition；stationarity非充分；rank deficient是不適用而非無極值，且本條件不是inequality KKT、global或uniqueness證書。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Lagrange乘子條件"]},"variant":{"family_id":"qf-mod50-lagrange-regularity-audit","seed":202611194017,"parameters":{"constraint_gradients":[[1,0],[0,17]],"domain":"Exact_equality_constraint_Lagrange_condition","operation":"lagrange-regularity-audit","semantic_template":"lagrange-regularity-audit|{\"constraint_gradients\":[[[0,1],[17,1]],[[1,1],[0,1]]]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r04-0018","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-multiplier-condition"],"title_zh":"Lagrange乘子條件","stem_zh":"feasibility、constraint-gradient full row rank與grad_f=J_g^T*lambda convention。操作=lagrange-regularity-audit；明示資料={\"constraint_gradients\":[[1,0],[0,18]]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 18]], 2, 2, 2, 是, full_row_rank_means_constraint_gradients_are_linearly_independent)","explanation_zh":"Wave6第十三層核對=([[1, 0], [0, 18]], 2, 2, 2, 是, full_row_rank_means_constraint_gradients_are_linearly_independent)；依完整明示資料精確重算；regular equality-constrained local extremum只得到必要multiplier condition；stationarity非充分；rank deficient是不適用而非無極值，且本條件不是inequality KKT、global或uniqueness證書。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Lagrange乘子條件"]},"variant":{"family_id":"qf-mod50-lagrange-regularity-audit","seed":202611194018,"parameters":{"constraint_gradients":[[1,0],[0,18]],"domain":"Exact_equality_constraint_Lagrange_condition","operation":"lagrange-regularity-audit","semantic_template":"lagrange-regularity-audit|{\"constraint_gradients\":[[[0,1],[18,1]],[[1,1],[0,1]]]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r04-0019","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-multiplier-condition"],"title_zh":"Lagrange乘子條件","stem_zh":"feasibility、constraint-gradient full row rank與grad_f=J_g^T*lambda convention。操作=lagrange-regularity-audit；明示資料={\"constraint_gradients\":[[1,0],[0,19]]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 19]], 2, 2, 2, 是, full_row_rank_means_constraint_gradients_are_linearly_independent)","explanation_zh":"Wave6第十三層核對=([[1, 0], [0, 19]], 2, 2, 2, 是, full_row_rank_means_constraint_gradients_are_linearly_independent)；依完整明示資料精確重算；regular equality-constrained local extremum只得到必要multiplier condition；stationarity非充分；rank deficient是不適用而非無極值，且本條件不是inequality KKT、global或uniqueness證書。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Lagrange乘子條件"]},"variant":{"family_id":"qf-mod50-lagrange-regularity-audit","seed":202611194019,"parameters":{"constraint_gradients":[[1,0],[0,19]],"domain":"Exact_equality_constraint_Lagrange_condition","operation":"lagrange-regularity-audit","semantic_template":"lagrange-regularity-audit|{\"constraint_gradients\":[[[0,1],[19,1]],[[1,1],[0,1]]]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r04-0020","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-multiplier-condition"],"title_zh":"Lagrange乘子條件","stem_zh":"feasibility、constraint-gradient full row rank與grad_f=J_g^T*lambda convention。操作=lagrange-regularity-audit；明示資料={\"constraint_gradients\":[[1,0],[0,20]]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 20]], 2, 2, 2, 是, full_row_rank_means_constraint_gradients_are_linearly_independent)","explanation_zh":"Wave6第十三層核對=([[1, 0], [0, 20]], 2, 2, 2, 是, full_row_rank_means_constraint_gradients_are_linearly_independent)；依完整明示資料精確重算；regular equality-constrained local extremum只得到必要multiplier condition；stationarity非充分；rank deficient是不適用而非無極值，且本條件不是inequality KKT、global或uniqueness證書。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Lagrange乘子條件"]},"variant":{"family_id":"qf-mod50-lagrange-regularity-audit","seed":202611194020,"parameters":{"constraint_gradients":[[1,0],[0,20]],"domain":"Exact_equality_constraint_Lagrange_condition","operation":"lagrange-regularity-audit","semantic_template":"lagrange-regularity-audit|{\"constraint_gradients\":[[[0,1],[20,1]],[[1,1],[0,1]]]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r04-0021","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-multiplier-condition"],"title_zh":"Lagrange乘子條件","stem_zh":"feasibility、constraint-gradient full row rank與grad_f=J_g^T*lambda convention。操作=lagrange-regularity-audit；明示資料={\"constraint_gradients\":[[1,0],[0,21]]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 21]], 2, 2, 2, 是, full_row_rank_means_constraint_gradients_are_linearly_independent)","explanation_zh":"Wave6第十三層核對=([[1, 0], [0, 21]], 2, 2, 2, 是, full_row_rank_means_constraint_gradients_are_linearly_independent)；依完整明示資料精確重算；regular equality-constrained local extremum只得到必要multiplier condition；stationarity非充分；rank deficient是不適用而非無極值，且本條件不是inequality KKT、global或uniqueness證書。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Lagrange乘子條件"]},"variant":{"family_id":"qf-mod50-lagrange-regularity-audit","seed":202611194021,"parameters":{"constraint_gradients":[[1,0],[0,21]],"domain":"Exact_equality_constraint_Lagrange_condition","operation":"lagrange-regularity-audit","semantic_template":"lagrange-regularity-audit|{\"constraint_gradients\":[[[0,1],[21,1]],[[1,1],[0,1]]]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r04-0022","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-multiplier-condition"],"title_zh":"Lagrange乘子條件","stem_zh":"feasibility、constraint-gradient full row rank與grad_f=J_g^T*lambda convention。操作=lagrange-regularity-audit；明示資料={\"constraint_gradients\":[[1,0],[0,22]]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 22]], 2, 2, 2, 是, full_row_rank_means_constraint_gradients_are_linearly_independent)","explanation_zh":"Wave6第十三層核對=([[1, 0], [0, 22]], 2, 2, 2, 是, full_row_rank_means_constraint_gradients_are_linearly_independent)；依完整明示資料精確重算；regular equality-constrained local extremum只得到必要multiplier condition；stationarity非充分；rank deficient是不適用而非無極值，且本條件不是inequality KKT、global或uniqueness證書。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Lagrange乘子條件"]},"variant":{"family_id":"qf-mod50-lagrange-regularity-audit","seed":202611194022,"parameters":{"constraint_gradients":[[1,0],[0,22]],"domain":"Exact_equality_constraint_Lagrange_condition","operation":"lagrange-regularity-audit","semantic_template":"lagrange-regularity-audit|{\"constraint_gradients\":[[[0,1],[22,1]],[[1,1],[0,1]]]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r04-0023","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-multiplier-condition"],"title_zh":"Lagrange乘子條件","stem_zh":"feasibility、constraint-gradient full row rank與grad_f=J_g^T*lambda convention。操作=lagrange-regularity-audit；明示資料={\"constraint_gradients\":[[1,0],[0,23]]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 23]], 2, 2, 2, 是, full_row_rank_means_constraint_gradients_are_linearly_independent)","explanation_zh":"Wave6第十三層核對=([[1, 0], [0, 23]], 2, 2, 2, 是, full_row_rank_means_constraint_gradients_are_linearly_independent)；依完整明示資料精確重算；regular equality-constrained local extremum只得到必要multiplier condition；stationarity非充分；rank deficient是不適用而非無極值，且本條件不是inequality KKT、global或uniqueness證書。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Lagrange乘子條件"]},"variant":{"family_id":"qf-mod50-lagrange-regularity-audit","seed":202611194023,"parameters":{"constraint_gradients":[[1,0],[0,23]],"domain":"Exact_equality_constraint_Lagrange_condition","operation":"lagrange-regularity-audit","semantic_template":"lagrange-regularity-audit|{\"constraint_gradients\":[[[0,1],[23,1]],[[1,1],[0,1]]]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r04-0024","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-multiplier-condition"],"title_zh":"Lagrange乘子條件","stem_zh":"feasibility、constraint-gradient full row rank與grad_f=J_g^T*lambda convention。操作=lagrange-regularity-audit；明示資料={\"constraint_gradients\":[[1,0],[0,24]]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 24]], 2, 2, 2, 是, full_row_rank_means_constraint_gradients_are_linearly_independent)","explanation_zh":"Wave6第十三層核對=([[1, 0], [0, 24]], 2, 2, 2, 是, full_row_rank_means_constraint_gradients_are_linearly_independent)；依完整明示資料精確重算；regular equality-constrained local extremum只得到必要multiplier condition；stationarity非充分；rank deficient是不適用而非無極值，且本條件不是inequality KKT、global或uniqueness證書。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Lagrange乘子條件"]},"variant":{"family_id":"qf-mod50-lagrange-regularity-audit","seed":202611194024,"parameters":{"constraint_gradients":[[1,0],[0,24]],"domain":"Exact_equality_constraint_Lagrange_condition","operation":"lagrange-regularity-audit","semantic_template":"lagrange-regularity-audit|{\"constraint_gradients\":[[[0,1],[24,1]],[[1,1],[0,1]]]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r04-0025","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-multiplier-condition"],"title_zh":"Lagrange乘子條件","stem_zh":"feasibility、constraint-gradient full row rank與grad_f=J_g^T*lambda convention。操作=lagrange-regularity-audit；明示資料={\"constraint_gradients\":[[1,0],[0,25]]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([[1, 0], [0, 25]], 2, 2, 2, 是, full_row_rank_means_constraint_gradients_are_linearly_independent)","explanation_zh":"Wave6第十三層核對=([[1, 0], [0, 25]], 2, 2, 2, 是, full_row_rank_means_constraint_gradients_are_linearly_independent)；依完整明示資料精確重算；regular equality-constrained local extremum只得到必要multiplier condition；stationarity非充分；rank deficient是不適用而非無極值，且本條件不是inequality KKT、global或uniqueness證書。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Lagrange乘子條件"]},"variant":{"family_id":"qf-mod50-lagrange-regularity-audit","seed":202611194025,"parameters":{"constraint_gradients":[[1,0],[0,25]],"domain":"Exact_equality_constraint_Lagrange_condition","operation":"lagrange-regularity-audit","semantic_template":"lagrange-regularity-audit|{\"constraint_gradients\":[[[0,1],[25,1]],[[1,1],[0,1]]]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r04-0026","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-multiplier-condition"],"title_zh":"Lagrange乘子條件","stem_zh":"feasibility、constraint-gradient full row rank與grad_f=J_g^T*lambda convention。操作=lagrange-multiplier-residual；明示資料={\"constraint_gradients\":[[1,0],[0,1]],\"multipliers\":[1,2],\"objective_gradient\":[1,2]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([1, 2], [[1, 0], [0, 1]], [1, 2], [1, 2], [0, 0], 是, convention_grad_f_equals_J_g_transpose_lambda)","explanation_zh":"Wave6第十三層核對=([1, 2], [[1, 0], [0, 1]], [1, 2], [1, 2], [0, 0], 是, convention_grad_f_equals_J_g_transpose_lambda)；依完整明示資料精確重算；regular equality-constrained local extremum只得到必要multiplier condition；stationarity非充分；rank deficient是不適用而非無極值，且本條件不是inequality KKT、global或uniqueness證書。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Lagrange乘子條件"]},"variant":{"family_id":"qf-mod50-lagrange-multiplier-residual","seed":202611194026,"parameters":{"objective_gradient":[1,2],"constraint_gradients":[[1,0],[0,1]],"multipliers":[1,2],"domain":"Exact_equality_constraint_Lagrange_condition","operation":"lagrange-multiplier-residual","semantic_template":"lagrange-multiplier-residual|{\"constraint_gradients\":[[[0,1],[1,1]],[[1,1],[0,1]]],\"multipliers\":[[2,1],[1,1]],\"objective_gradient\":[[1,1],[2,1]]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r04-0027","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-multiplier-condition"],"title_zh":"Lagrange乘子條件","stem_zh":"feasibility、constraint-gradient full row rank與grad_f=J_g^T*lambda convention。操作=lagrange-multiplier-residual；明示資料={\"constraint_gradients\":[[1,0],[0,1]],\"multipliers\":[2,4],\"objective_gradient\":[2,4]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([2, 4], [[1, 0], [0, 1]], [2, 4], [2, 4], [0, 0], 是, convention_grad_f_equals_J_g_transpose_lambda)","explanation_zh":"Wave6第十三層核對=([2, 4], [[1, 0], [0, 1]], [2, 4], [2, 4], [0, 0], 是, convention_grad_f_equals_J_g_transpose_lambda)；依完整明示資料精確重算；regular equality-constrained local extremum只得到必要multiplier condition；stationarity非充分；rank deficient是不適用而非無極值，且本條件不是inequality KKT、global或uniqueness證書。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Lagrange乘子條件"]},"variant":{"family_id":"qf-mod50-lagrange-multiplier-residual","seed":202611194027,"parameters":{"objective_gradient":[2,4],"constraint_gradients":[[1,0],[0,1]],"multipliers":[2,4],"domain":"Exact_equality_constraint_Lagrange_condition","operation":"lagrange-multiplier-residual","semantic_template":"lagrange-multiplier-residual|{\"constraint_gradients\":[[[0,1],[1,1]],[[1,1],[0,1]]],\"multipliers\":[[4,1],[2,1]],\"objective_gradient\":[[2,1],[4,1]]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r04-0028","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-multiplier-condition"],"title_zh":"Lagrange乘子條件","stem_zh":"feasibility、constraint-gradient full row rank與grad_f=J_g^T*lambda convention。操作=lagrange-multiplier-residual；明示資料={\"constraint_gradients\":[[1,0],[0,1]],\"multipliers\":[3,6],\"objective_gradient\":[3,6]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([3, 6], [[1, 0], [0, 1]], [3, 6], [3, 6], [0, 0], 是, convention_grad_f_equals_J_g_transpose_lambda)","explanation_zh":"Wave6第十三層核對=([3, 6], [[1, 0], [0, 1]], [3, 6], [3, 6], [0, 0], 是, convention_grad_f_equals_J_g_transpose_lambda)；依完整明示資料精確重算；regular equality-constrained local extremum只得到必要multiplier condition；stationarity非充分；rank deficient是不適用而非無極值，且本條件不是inequality KKT、global或uniqueness證書。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Lagrange乘子條件"]},"variant":{"family_id":"qf-mod50-lagrange-multiplier-residual","seed":202611194028,"parameters":{"objective_gradient":[3,6],"constraint_gradients":[[1,0],[0,1]],"multipliers":[3,6],"domain":"Exact_equality_constraint_Lagrange_condition","operation":"lagrange-multiplier-residual","semantic_template":"lagrange-multiplier-residual|{\"constraint_gradients\":[[[0,1],[1,1]],[[1,1],[0,1]]],\"multipliers\":[[6,1],[3,1]],\"objective_gradient\":[[3,1],[6,1]]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r04-0029","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-multiplier-condition"],"title_zh":"Lagrange乘子條件","stem_zh":"feasibility、constraint-gradient full row rank與grad_f=J_g^T*lambda convention。操作=lagrange-multiplier-residual；明示資料={\"constraint_gradients\":[[1,0],[0,1]],\"multipliers\":[4,8],\"objective_gradient\":[4,8]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([4, 8], [[1, 0], [0, 1]], [4, 8], [4, 8], [0, 0], 是, convention_grad_f_equals_J_g_transpose_lambda)","explanation_zh":"Wave6第十三層核對=([4, 8], [[1, 0], [0, 1]], [4, 8], [4, 8], [0, 0], 是, convention_grad_f_equals_J_g_transpose_lambda)；依完整明示資料精確重算；regular equality-constrained local extremum只得到必要multiplier condition；stationarity非充分；rank deficient是不適用而非無極值，且本條件不是inequality KKT、global或uniqueness證書。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Lagrange乘子條件"]},"variant":{"family_id":"qf-mod50-lagrange-multiplier-residual","seed":202611194029,"parameters":{"objective_gradient":[4,8],"constraint_gradients":[[1,0],[0,1]],"multipliers":[4,8],"domain":"Exact_equality_constraint_Lagrange_condition","operation":"lagrange-multiplier-residual","semantic_template":"lagrange-multiplier-residual|{\"constraint_gradients\":[[[0,1],[1,1]],[[1,1],[0,1]]],\"multipliers\":[[8,1],[4,1]],\"objective_gradient\":[[4,1],[8,1]]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r04-0030","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-multiplier-condition"],"title_zh":"Lagrange乘子條件","stem_zh":"feasibility、constraint-gradient full row rank與grad_f=J_g^T*lambda convention。操作=lagrange-multiplier-residual；明示資料={\"constraint_gradients\":[[1,0],[0,1]],\"multipliers\":[5,10],\"objective_gradient\":[5,10]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([5, 10], [[1, 0], [0, 1]], [5, 10], [5, 10], [0, 0], 是, convention_grad_f_equals_J_g_transpose_lambda)","explanation_zh":"Wave6第十三層核對=([5, 10], [[1, 0], [0, 1]], [5, 10], [5, 10], [0, 0], 是, convention_grad_f_equals_J_g_transpose_lambda)；依完整明示資料精確重算；regular equality-constrained local extremum只得到必要multiplier condition；stationarity非充分；rank deficient是不適用而非無極值，且本條件不是inequality KKT、global或uniqueness證書。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Lagrange乘子條件"]},"variant":{"family_id":"qf-mod50-lagrange-multiplier-residual","seed":202611194030,"parameters":{"objective_gradient":[5,10],"constraint_gradients":[[1,0],[0,1]],"multipliers":[5,10],"domain":"Exact_equality_constraint_Lagrange_condition","operation":"lagrange-multiplier-residual","semantic_template":"lagrange-multiplier-residual|{\"constraint_gradients\":[[[0,1],[1,1]],[[1,1],[0,1]]],\"multipliers\":[[10,1],[5,1]],\"objective_gradient\":[[5,1],[10,1]]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r04-0031","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-multiplier-condition"],"title_zh":"Lagrange乘子條件","stem_zh":"feasibility、constraint-gradient full row rank與grad_f=J_g^T*lambda convention。操作=lagrange-multiplier-residual；明示資料={\"constraint_gradients\":[[1,0],[0,1]],\"multipliers\":[6,12],\"objective_gradient\":[6,12]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([6, 12], [[1, 0], [0, 1]], [6, 12], [6, 12], [0, 0], 是, convention_grad_f_equals_J_g_transpose_lambda)","explanation_zh":"Wave6第十三層核對=([6, 12], [[1, 0], [0, 1]], [6, 12], [6, 12], [0, 0], 是, convention_grad_f_equals_J_g_transpose_lambda)；依完整明示資料精確重算；regular equality-constrained local extremum只得到必要multiplier condition；stationarity非充分；rank deficient是不適用而非無極值，且本條件不是inequality KKT、global或uniqueness證書。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Lagrange乘子條件"]},"variant":{"family_id":"qf-mod50-lagrange-multiplier-residual","seed":202611194031,"parameters":{"objective_gradient":[6,12],"constraint_gradients":[[1,0],[0,1]],"multipliers":[6,12],"domain":"Exact_equality_constraint_Lagrange_condition","operation":"lagrange-multiplier-residual","semantic_template":"lagrange-multiplier-residual|{\"constraint_gradients\":[[[0,1],[1,1]],[[1,1],[0,1]]],\"multipliers\":[[12,1],[6,1]],\"objective_gradient\":[[6,1],[12,1]]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r04-0032","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-multiplier-condition"],"title_zh":"Lagrange乘子條件","stem_zh":"feasibility、constraint-gradient full row rank與grad_f=J_g^T*lambda convention。操作=lagrange-multiplier-residual；明示資料={\"constraint_gradients\":[[1,0],[0,1]],\"multipliers\":[7,14],\"objective_gradient\":[7,14]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([7, 14], [[1, 0], [0, 1]], [7, 14], [7, 14], [0, 0], 是, convention_grad_f_equals_J_g_transpose_lambda)","explanation_zh":"Wave6第十三層核對=([7, 14], [[1, 0], [0, 1]], [7, 14], [7, 14], [0, 0], 是, convention_grad_f_equals_J_g_transpose_lambda)；依完整明示資料精確重算；regular equality-constrained local extremum只得到必要multiplier condition；stationarity非充分；rank deficient是不適用而非無極值，且本條件不是inequality KKT、global或uniqueness證書。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Lagrange乘子條件"]},"variant":{"family_id":"qf-mod50-lagrange-multiplier-residual","seed":202611194032,"parameters":{"objective_gradient":[7,14],"constraint_gradients":[[1,0],[0,1]],"multipliers":[7,14],"domain":"Exact_equality_constraint_Lagrange_condition","operation":"lagrange-multiplier-residual","semantic_template":"lagrange-multiplier-residual|{\"constraint_gradients\":[[[0,1],[1,1]],[[1,1],[0,1]]],\"multipliers\":[[14,1],[7,1]],\"objective_gradient\":[[7,1],[14,1]]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r04-0033","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-multiplier-condition"],"title_zh":"Lagrange乘子條件","stem_zh":"feasibility、constraint-gradient full row rank與grad_f=J_g^T*lambda convention。操作=lagrange-multiplier-residual；明示資料={\"constraint_gradients\":[[1,0],[0,1]],\"multipliers\":[8,16],\"objective_gradient\":[8,16]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([8, 16], [[1, 0], [0, 1]], [8, 16], [8, 16], [0, 0], 是, convention_grad_f_equals_J_g_transpose_lambda)","explanation_zh":"Wave6第十三層核對=([8, 16], [[1, 0], [0, 1]], [8, 16], [8, 16], [0, 0], 是, convention_grad_f_equals_J_g_transpose_lambda)；依完整明示資料精確重算；regular equality-constrained local extremum只得到必要multiplier condition；stationarity非充分；rank deficient是不適用而非無極值，且本條件不是inequality KKT、global或uniqueness證書。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Lagrange乘子條件"]},"variant":{"family_id":"qf-mod50-lagrange-multiplier-residual","seed":202611194033,"parameters":{"objective_gradient":[8,16],"constraint_gradients":[[1,0],[0,1]],"multipliers":[8,16],"domain":"Exact_equality_constraint_Lagrange_condition","operation":"lagrange-multiplier-residual","semantic_template":"lagrange-multiplier-residual|{\"constraint_gradients\":[[[0,1],[1,1]],[[1,1],[0,1]]],\"multipliers\":[[16,1],[8,1]],\"objective_gradient\":[[8,1],[16,1]]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r04-0034","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-multiplier-condition"],"title_zh":"Lagrange乘子條件","stem_zh":"feasibility、constraint-gradient full row rank與grad_f=J_g^T*lambda convention。操作=lagrange-multiplier-residual；明示資料={\"constraint_gradients\":[[1,0],[0,1]],\"multipliers\":[9,18],\"objective_gradient\":[9,18]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([9, 18], [[1, 0], [0, 1]], [9, 18], [9, 18], [0, 0], 是, convention_grad_f_equals_J_g_transpose_lambda)","explanation_zh":"Wave6第十三層核對=([9, 18], [[1, 0], [0, 1]], [9, 18], [9, 18], [0, 0], 是, convention_grad_f_equals_J_g_transpose_lambda)；依完整明示資料精確重算；regular equality-constrained local extremum只得到必要multiplier condition；stationarity非充分；rank deficient是不適用而非無極值，且本條件不是inequality KKT、global或uniqueness證書。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Lagrange乘子條件"]},"variant":{"family_id":"qf-mod50-lagrange-multiplier-residual","seed":202611194034,"parameters":{"objective_gradient":[9,18],"constraint_gradients":[[1,0],[0,1]],"multipliers":[9,18],"domain":"Exact_equality_constraint_Lagrange_condition","operation":"lagrange-multiplier-residual","semantic_template":"lagrange-multiplier-residual|{\"constraint_gradients\":[[[0,1],[1,1]],[[1,1],[0,1]]],\"multipliers\":[[18,1],[9,1]],\"objective_gradient\":[[9,1],[18,1]]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r04-0035","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-multiplier-condition"],"title_zh":"Lagrange乘子條件","stem_zh":"feasibility、constraint-gradient full row rank與grad_f=J_g^T*lambda convention。操作=lagrange-multiplier-residual；明示資料={\"constraint_gradients\":[[1,0],[0,1]],\"multipliers\":[10,20],\"objective_gradient\":[10,20]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([10, 20], [[1, 0], [0, 1]], [10, 20], [10, 20], [0, 0], 是, convention_grad_f_equals_J_g_transpose_lambda)","explanation_zh":"Wave6第十三層核對=([10, 20], [[1, 0], [0, 1]], [10, 20], [10, 20], [0, 0], 是, convention_grad_f_equals_J_g_transpose_lambda)；依完整明示資料精確重算；regular equality-constrained local extremum只得到必要multiplier condition；stationarity非充分；rank deficient是不適用而非無極值，且本條件不是inequality KKT、global或uniqueness證書。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Lagrange乘子條件"]},"variant":{"family_id":"qf-mod50-lagrange-multiplier-residual","seed":202611194035,"parameters":{"objective_gradient":[10,20],"constraint_gradients":[[1,0],[0,1]],"multipliers":[10,20],"domain":"Exact_equality_constraint_Lagrange_condition","operation":"lagrange-multiplier-residual","semantic_template":"lagrange-multiplier-residual|{\"constraint_gradients\":[[[0,1],[1,1]],[[1,1],[0,1]]],\"multipliers\":[[20,1],[10,1]],\"objective_gradient\":[[10,1],[20,1]]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r04-0036","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-multiplier-condition"],"title_zh":"Lagrange乘子條件","stem_zh":"feasibility、constraint-gradient full row rank與grad_f=J_g^T*lambda convention。操作=lagrange-multiplier-residual；明示資料={\"constraint_gradients\":[[1,0],[0,1]],\"multipliers\":[11,22],\"objective_gradient\":[11,22]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([11, 22], [[1, 0], [0, 1]], [11, 22], [11, 22], [0, 0], 是, convention_grad_f_equals_J_g_transpose_lambda)","explanation_zh":"Wave6第十三層核對=([11, 22], [[1, 0], [0, 1]], [11, 22], [11, 22], [0, 0], 是, convention_grad_f_equals_J_g_transpose_lambda)；依完整明示資料精確重算；regular equality-constrained local extremum只得到必要multiplier condition；stationarity非充分；rank deficient是不適用而非無極值，且本條件不是inequality KKT、global或uniqueness證書。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Lagrange乘子條件"]},"variant":{"family_id":"qf-mod50-lagrange-multiplier-residual","seed":202611194036,"parameters":{"objective_gradient":[11,22],"constraint_gradients":[[1,0],[0,1]],"multipliers":[11,22],"domain":"Exact_equality_constraint_Lagrange_condition","operation":"lagrange-multiplier-residual","semantic_template":"lagrange-multiplier-residual|{\"constraint_gradients\":[[[0,1],[1,1]],[[1,1],[0,1]]],\"multipliers\":[[22,1],[11,1]],\"objective_gradient\":[[11,1],[22,1]]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r04-0037","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-multiplier-condition"],"title_zh":"Lagrange乘子條件","stem_zh":"feasibility、constraint-gradient full row rank與grad_f=J_g^T*lambda convention。操作=lagrange-multiplier-residual；明示資料={\"constraint_gradients\":[[1,0],[0,1]],\"multipliers\":[12,24],\"objective_gradient\":[12,24]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([12, 24], [[1, 0], [0, 1]], [12, 24], [12, 24], [0, 0], 是, convention_grad_f_equals_J_g_transpose_lambda)","explanation_zh":"Wave6第十三層核對=([12, 24], [[1, 0], [0, 1]], [12, 24], [12, 24], [0, 0], 是, convention_grad_f_equals_J_g_transpose_lambda)；依完整明示資料精確重算；regular equality-constrained local extremum只得到必要multiplier condition；stationarity非充分；rank deficient是不適用而非無極值，且本條件不是inequality KKT、global或uniqueness證書。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Lagrange乘子條件"]},"variant":{"family_id":"qf-mod50-lagrange-multiplier-residual","seed":202611194037,"parameters":{"objective_gradient":[12,24],"constraint_gradients":[[1,0],[0,1]],"multipliers":[12,24],"domain":"Exact_equality_constraint_Lagrange_condition","operation":"lagrange-multiplier-residual","semantic_template":"lagrange-multiplier-residual|{\"constraint_gradients\":[[[0,1],[1,1]],[[1,1],[0,1]]],\"multipliers\":[[24,1],[12,1]],\"objective_gradient\":[[12,1],[24,1]]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r04-0038","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-multiplier-condition"],"title_zh":"Lagrange乘子條件","stem_zh":"feasibility、constraint-gradient full row rank與grad_f=J_g^T*lambda convention。操作=lagrange-multiplier-residual；明示資料={\"constraint_gradients\":[[1,0],[0,1]],\"multipliers\":[13,26],\"objective_gradient\":[13,26]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([13, 26], [[1, 0], [0, 1]], [13, 26], [13, 26], [0, 0], 是, convention_grad_f_equals_J_g_transpose_lambda)","explanation_zh":"Wave6第十三層核對=([13, 26], [[1, 0], [0, 1]], [13, 26], [13, 26], [0, 0], 是, convention_grad_f_equals_J_g_transpose_lambda)；依完整明示資料精確重算；regular equality-constrained local extremum只得到必要multiplier condition；stationarity非充分；rank deficient是不適用而非無極值，且本條件不是inequality KKT、global或uniqueness證書。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Lagrange乘子條件"]},"variant":{"family_id":"qf-mod50-lagrange-multiplier-residual","seed":202611194038,"parameters":{"objective_gradient":[13,26],"constraint_gradients":[[1,0],[0,1]],"multipliers":[13,26],"domain":"Exact_equality_constraint_Lagrange_condition","operation":"lagrange-multiplier-residual","semantic_template":"lagrange-multiplier-residual|{\"constraint_gradients\":[[[0,1],[1,1]],[[1,1],[0,1]]],\"multipliers\":[[26,1],[13,1]],\"objective_gradient\":[[13,1],[26,1]]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r04-0039","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-multiplier-condition"],"title_zh":"Lagrange乘子條件","stem_zh":"feasibility、constraint-gradient full row rank與grad_f=J_g^T*lambda convention。操作=lagrange-multiplier-residual；明示資料={\"constraint_gradients\":[[1,0],[0,1]],\"multipliers\":[14,28],\"objective_gradient\":[14,28]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([14, 28], [[1, 0], [0, 1]], [14, 28], [14, 28], [0, 0], 是, convention_grad_f_equals_J_g_transpose_lambda)","explanation_zh":"Wave6第十三層核對=([14, 28], [[1, 0], [0, 1]], [14, 28], [14, 28], [0, 0], 是, convention_grad_f_equals_J_g_transpose_lambda)；依完整明示資料精確重算；regular equality-constrained local extremum只得到必要multiplier condition；stationarity非充分；rank deficient是不適用而非無極值，且本條件不是inequality KKT、global或uniqueness證書。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Lagrange乘子條件"]},"variant":{"family_id":"qf-mod50-lagrange-multiplier-residual","seed":202611194039,"parameters":{"objective_gradient":[14,28],"constraint_gradients":[[1,0],[0,1]],"multipliers":[14,28],"domain":"Exact_equality_constraint_Lagrange_condition","operation":"lagrange-multiplier-residual","semantic_template":"lagrange-multiplier-residual|{\"constraint_gradients\":[[[0,1],[1,1]],[[1,1],[0,1]]],\"multipliers\":[[28,1],[14,1]],\"objective_gradient\":[[14,1],[28,1]]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r04-0040","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-multiplier-condition"],"title_zh":"Lagrange乘子條件","stem_zh":"feasibility、constraint-gradient full row rank與grad_f=J_g^T*lambda convention。操作=lagrange-multiplier-residual；明示資料={\"constraint_gradients\":[[1,0],[0,1]],\"multipliers\":[15,30],\"objective_gradient\":[15,30]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([15, 30], [[1, 0], [0, 1]], [15, 30], [15, 30], [0, 0], 是, convention_grad_f_equals_J_g_transpose_lambda)","explanation_zh":"Wave6第十三層核對=([15, 30], [[1, 0], [0, 1]], [15, 30], [15, 30], [0, 0], 是, convention_grad_f_equals_J_g_transpose_lambda)；依完整明示資料精確重算；regular equality-constrained local extremum只得到必要multiplier condition；stationarity非充分；rank deficient是不適用而非無極值，且本條件不是inequality KKT、global或uniqueness證書。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Lagrange乘子條件"]},"variant":{"family_id":"qf-mod50-lagrange-multiplier-residual","seed":202611194040,"parameters":{"objective_gradient":[15,30],"constraint_gradients":[[1,0],[0,1]],"multipliers":[15,30],"domain":"Exact_equality_constraint_Lagrange_condition","operation":"lagrange-multiplier-residual","semantic_template":"lagrange-multiplier-residual|{\"constraint_gradients\":[[[0,1],[1,1]],[[1,1],[0,1]]],\"multipliers\":[[30,1],[15,1]],\"objective_gradient\":[[15,1],[30,1]]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r04-0041","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-multiplier-condition"],"title_zh":"Lagrange乘子條件","stem_zh":"feasibility、constraint-gradient full row rank與grad_f=J_g^T*lambda convention。操作=lagrange-multiplier-residual；明示資料={\"constraint_gradients\":[[1,0],[0,1]],\"multipliers\":[16,32],\"objective_gradient\":[16,32]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([16, 32], [[1, 0], [0, 1]], [16, 32], [16, 32], [0, 0], 是, convention_grad_f_equals_J_g_transpose_lambda)","explanation_zh":"Wave6第十三層核對=([16, 32], [[1, 0], [0, 1]], [16, 32], [16, 32], [0, 0], 是, convention_grad_f_equals_J_g_transpose_lambda)；依完整明示資料精確重算；regular equality-constrained local extremum只得到必要multiplier condition；stationarity非充分；rank deficient是不適用而非無極值，且本條件不是inequality KKT、global或uniqueness證書。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Lagrange乘子條件"]},"variant":{"family_id":"qf-mod50-lagrange-multiplier-residual","seed":202611194041,"parameters":{"objective_gradient":[16,32],"constraint_gradients":[[1,0],[0,1]],"multipliers":[16,32],"domain":"Exact_equality_constraint_Lagrange_condition","operation":"lagrange-multiplier-residual","semantic_template":"lagrange-multiplier-residual|{\"constraint_gradients\":[[[0,1],[1,1]],[[1,1],[0,1]]],\"multipliers\":[[32,1],[16,1]],\"objective_gradient\":[[16,1],[32,1]]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r04-0042","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-multiplier-condition"],"title_zh":"Lagrange乘子條件","stem_zh":"feasibility、constraint-gradient full row rank與grad_f=J_g^T*lambda convention。操作=lagrange-multiplier-residual；明示資料={\"constraint_gradients\":[[1,0],[0,1]],\"multipliers\":[17,34],\"objective_gradient\":[17,34]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([17, 34], [[1, 0], [0, 1]], [17, 34], [17, 34], [0, 0], 是, convention_grad_f_equals_J_g_transpose_lambda)","explanation_zh":"Wave6第十三層核對=([17, 34], [[1, 0], [0, 1]], [17, 34], [17, 34], [0, 0], 是, convention_grad_f_equals_J_g_transpose_lambda)；依完整明示資料精確重算；regular equality-constrained local extremum只得到必要multiplier condition；stationarity非充分；rank deficient是不適用而非無極值，且本條件不是inequality KKT、global或uniqueness證書。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Lagrange乘子條件"]},"variant":{"family_id":"qf-mod50-lagrange-multiplier-residual","seed":202611194042,"parameters":{"objective_gradient":[17,34],"constraint_gradients":[[1,0],[0,1]],"multipliers":[17,34],"domain":"Exact_equality_constraint_Lagrange_condition","operation":"lagrange-multiplier-residual","semantic_template":"lagrange-multiplier-residual|{\"constraint_gradients\":[[[0,1],[1,1]],[[1,1],[0,1]]],\"multipliers\":[[34,1],[17,1]],\"objective_gradient\":[[17,1],[34,1]]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r04-0043","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-multiplier-condition"],"title_zh":"Lagrange乘子條件","stem_zh":"feasibility、constraint-gradient full row rank與grad_f=J_g^T*lambda convention。操作=lagrange-multiplier-residual；明示資料={\"constraint_gradients\":[[1,0],[0,1]],\"multipliers\":[18,36],\"objective_gradient\":[18,36]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([18, 36], [[1, 0], [0, 1]], [18, 36], [18, 36], [0, 0], 是, convention_grad_f_equals_J_g_transpose_lambda)","explanation_zh":"Wave6第十三層核對=([18, 36], [[1, 0], [0, 1]], [18, 36], [18, 36], [0, 0], 是, convention_grad_f_equals_J_g_transpose_lambda)；依完整明示資料精確重算；regular equality-constrained local extremum只得到必要multiplier condition；stationarity非充分；rank deficient是不適用而非無極值，且本條件不是inequality KKT、global或uniqueness證書。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Lagrange乘子條件"]},"variant":{"family_id":"qf-mod50-lagrange-multiplier-residual","seed":202611194043,"parameters":{"objective_gradient":[18,36],"constraint_gradients":[[1,0],[0,1]],"multipliers":[18,36],"domain":"Exact_equality_constraint_Lagrange_condition","operation":"lagrange-multiplier-residual","semantic_template":"lagrange-multiplier-residual|{\"constraint_gradients\":[[[0,1],[1,1]],[[1,1],[0,1]]],\"multipliers\":[[36,1],[18,1]],\"objective_gradient\":[[18,1],[36,1]]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r04-0044","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-multiplier-condition"],"title_zh":"Lagrange乘子條件","stem_zh":"feasibility、constraint-gradient full row rank與grad_f=J_g^T*lambda convention。操作=lagrange-multiplier-residual；明示資料={\"constraint_gradients\":[[1,0],[0,1]],\"multipliers\":[19,38],\"objective_gradient\":[19,38]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([19, 38], [[1, 0], [0, 1]], [19, 38], [19, 38], [0, 0], 是, convention_grad_f_equals_J_g_transpose_lambda)","explanation_zh":"Wave6第十三層核對=([19, 38], [[1, 0], [0, 1]], [19, 38], [19, 38], [0, 0], 是, convention_grad_f_equals_J_g_transpose_lambda)；依完整明示資料精確重算；regular equality-constrained local extremum只得到必要multiplier condition；stationarity非充分；rank deficient是不適用而非無極值，且本條件不是inequality KKT、global或uniqueness證書。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Lagrange乘子條件"]},"variant":{"family_id":"qf-mod50-lagrange-multiplier-residual","seed":202611194044,"parameters":{"objective_gradient":[19,38],"constraint_gradients":[[1,0],[0,1]],"multipliers":[19,38],"domain":"Exact_equality_constraint_Lagrange_condition","operation":"lagrange-multiplier-residual","semantic_template":"lagrange-multiplier-residual|{\"constraint_gradients\":[[[0,1],[1,1]],[[1,1],[0,1]]],\"multipliers\":[[38,1],[19,1]],\"objective_gradient\":[[19,1],[38,1]]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r04-0045","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-multiplier-condition"],"title_zh":"Lagrange乘子條件","stem_zh":"feasibility、constraint-gradient full row rank與grad_f=J_g^T*lambda convention。操作=lagrange-multiplier-residual；明示資料={\"constraint_gradients\":[[1,0],[0,1]],\"multipliers\":[20,40],\"objective_gradient\":[20,40]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([20, 40], [[1, 0], [0, 1]], [20, 40], [20, 40], [0, 0], 是, convention_grad_f_equals_J_g_transpose_lambda)","explanation_zh":"Wave6第十三層核對=([20, 40], [[1, 0], [0, 1]], [20, 40], [20, 40], [0, 0], 是, convention_grad_f_equals_J_g_transpose_lambda)；依完整明示資料精確重算；regular equality-constrained local extremum只得到必要multiplier condition；stationarity非充分；rank deficient是不適用而非無極值，且本條件不是inequality KKT、global或uniqueness證書。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Lagrange乘子條件"]},"variant":{"family_id":"qf-mod50-lagrange-multiplier-residual","seed":202611194045,"parameters":{"objective_gradient":[20,40],"constraint_gradients":[[1,0],[0,1]],"multipliers":[20,40],"domain":"Exact_equality_constraint_Lagrange_condition","operation":"lagrange-multiplier-residual","semantic_template":"lagrange-multiplier-residual|{\"constraint_gradients\":[[[0,1],[1,1]],[[1,1],[0,1]]],\"multipliers\":[[40,1],[20,1]],\"objective_gradient\":[[20,1],[40,1]]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r04-0046","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-multiplier-condition"],"title_zh":"Lagrange乘子條件","stem_zh":"feasibility、constraint-gradient full row rank與grad_f=J_g^T*lambda convention。操作=lagrange-multiplier-residual；明示資料={\"constraint_gradients\":[[1,0],[0,1]],\"multipliers\":[21,42],\"objective_gradient\":[21,42]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([21, 42], [[1, 0], [0, 1]], [21, 42], [21, 42], [0, 0], 是, convention_grad_f_equals_J_g_transpose_lambda)","explanation_zh":"Wave6第十三層核對=([21, 42], [[1, 0], [0, 1]], [21, 42], [21, 42], [0, 0], 是, convention_grad_f_equals_J_g_transpose_lambda)；依完整明示資料精確重算；regular equality-constrained local extremum只得到必要multiplier condition；stationarity非充分；rank deficient是不適用而非無極值，且本條件不是inequality KKT、global或uniqueness證書。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Lagrange乘子條件"]},"variant":{"family_id":"qf-mod50-lagrange-multiplier-residual","seed":202611194046,"parameters":{"objective_gradient":[21,42],"constraint_gradients":[[1,0],[0,1]],"multipliers":[21,42],"domain":"Exact_equality_constraint_Lagrange_condition","operation":"lagrange-multiplier-residual","semantic_template":"lagrange-multiplier-residual|{\"constraint_gradients\":[[[0,1],[1,1]],[[1,1],[0,1]]],\"multipliers\":[[42,1],[21,1]],\"objective_gradient\":[[21,1],[42,1]]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r04-0047","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-multiplier-condition"],"title_zh":"Lagrange乘子條件","stem_zh":"feasibility、constraint-gradient full row rank與grad_f=J_g^T*lambda convention。操作=lagrange-multiplier-residual；明示資料={\"constraint_gradients\":[[1,0],[0,1]],\"multipliers\":[22,44],\"objective_gradient\":[22,44]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([22, 44], [[1, 0], [0, 1]], [22, 44], [22, 44], [0, 0], 是, convention_grad_f_equals_J_g_transpose_lambda)","explanation_zh":"Wave6第十三層核對=([22, 44], [[1, 0], [0, 1]], [22, 44], [22, 44], [0, 0], 是, convention_grad_f_equals_J_g_transpose_lambda)；依完整明示資料精確重算；regular equality-constrained local extremum只得到必要multiplier condition；stationarity非充分；rank deficient是不適用而非無極值，且本條件不是inequality KKT、global或uniqueness證書。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Lagrange乘子條件"]},"variant":{"family_id":"qf-mod50-lagrange-multiplier-residual","seed":202611194047,"parameters":{"objective_gradient":[22,44],"constraint_gradients":[[1,0],[0,1]],"multipliers":[22,44],"domain":"Exact_equality_constraint_Lagrange_condition","operation":"lagrange-multiplier-residual","semantic_template":"lagrange-multiplier-residual|{\"constraint_gradients\":[[[0,1],[1,1]],[[1,1],[0,1]]],\"multipliers\":[[44,1],[22,1]],\"objective_gradient\":[[22,1],[44,1]]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r04-0048","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-multiplier-condition"],"title_zh":"Lagrange乘子條件","stem_zh":"feasibility、constraint-gradient full row rank與grad_f=J_g^T*lambda convention。操作=lagrange-multiplier-residual；明示資料={\"constraint_gradients\":[[1,0],[0,1]],\"multipliers\":[23,46],\"objective_gradient\":[23,46]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([23, 46], [[1, 0], [0, 1]], [23, 46], [23, 46], [0, 0], 是, convention_grad_f_equals_J_g_transpose_lambda)","explanation_zh":"Wave6第十三層核對=([23, 46], [[1, 0], [0, 1]], [23, 46], [23, 46], [0, 0], 是, convention_grad_f_equals_J_g_transpose_lambda)；依完整明示資料精確重算；regular equality-constrained local extremum只得到必要multiplier condition；stationarity非充分；rank deficient是不適用而非無極值，且本條件不是inequality KKT、global或uniqueness證書。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Lagrange乘子條件"]},"variant":{"family_id":"qf-mod50-lagrange-multiplier-residual","seed":202611194048,"parameters":{"objective_gradient":[23,46],"constraint_gradients":[[1,0],[0,1]],"multipliers":[23,46],"domain":"Exact_equality_constraint_Lagrange_condition","operation":"lagrange-multiplier-residual","semantic_template":"lagrange-multiplier-residual|{\"constraint_gradients\":[[[0,1],[1,1]],[[1,1],[0,1]]],\"multipliers\":[[46,1],[23,1]],\"objective_gradient\":[[23,1],[46,1]]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r04-0049","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-multiplier-condition"],"title_zh":"Lagrange乘子條件","stem_zh":"feasibility、constraint-gradient full row rank與grad_f=J_g^T*lambda convention。操作=lagrange-multiplier-residual；明示資料={\"constraint_gradients\":[[1,0],[0,1]],\"multipliers\":[24,48],\"objective_gradient\":[24,48]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([24, 48], [[1, 0], [0, 1]], [24, 48], [24, 48], [0, 0], 是, convention_grad_f_equals_J_g_transpose_lambda)","explanation_zh":"Wave6第十三層核對=([24, 48], [[1, 0], [0, 1]], [24, 48], [24, 48], [0, 0], 是, convention_grad_f_equals_J_g_transpose_lambda)；依完整明示資料精確重算；regular equality-constrained local extremum只得到必要multiplier condition；stationarity非充分；rank deficient是不適用而非無極值，且本條件不是inequality KKT、global或uniqueness證書。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Lagrange乘子條件"]},"variant":{"family_id":"qf-mod50-lagrange-multiplier-residual","seed":202611194049,"parameters":{"objective_gradient":[24,48],"constraint_gradients":[[1,0],[0,1]],"multipliers":[24,48],"domain":"Exact_equality_constraint_Lagrange_condition","operation":"lagrange-multiplier-residual","semantic_template":"lagrange-multiplier-residual|{\"constraint_gradients\":[[[0,1],[1,1]],[[1,1],[0,1]]],\"multipliers\":[[48,1],[24,1]],\"objective_gradient\":[[24,1],[48,1]]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r04-0050","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-multiplier-condition"],"title_zh":"Lagrange乘子條件","stem_zh":"feasibility、constraint-gradient full row rank與grad_f=J_g^T*lambda convention。操作=lagrange-multiplier-residual；明示資料={\"constraint_gradients\":[[1,0],[0,1]],\"multipliers\":[25,50],\"objective_gradient\":[25,50]}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([25, 50], [[1, 0], [0, 1]], [25, 50], [25, 50], [0, 0], 是, convention_grad_f_equals_J_g_transpose_lambda)","explanation_zh":"Wave6第十三層核對=([25, 50], [[1, 0], [0, 1]], [25, 50], [25, 50], [0, 0], 是, convention_grad_f_equals_J_g_transpose_lambda)；依完整明示資料精確重算；regular equality-constrained local extremum只得到必要multiplier condition；stationarity非充分；rank deficient是不適用而非無極值，且本條件不是inequality KKT、global或uniqueness證書。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Lagrange乘子條件"]},"variant":{"family_id":"qf-mod50-lagrange-multiplier-residual","seed":202611194050,"parameters":{"objective_gradient":[25,50],"constraint_gradients":[[1,0],[0,1]],"multipliers":[25,50],"domain":"Exact_equality_constraint_Lagrange_condition","operation":"lagrange-multiplier-residual","semantic_template":"lagrange-multiplier-residual|{\"constraint_gradients\":[[[0,1],[1,1]],[[1,1],[0,1]]],\"multipliers\":[[50,1],[25,1]],\"objective_gradient\":[[25,1],[50,1]]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r04-0051","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-multiplier-condition"],"title_zh":"Lagrange乘子條件","stem_zh":"feasibility、constraint-gradient full row rank與grad_f=J_g^T*lambda convention。操作=lagrange-single-linear-constraint；明示資料={\"center\":[0,0],\"constraint_gradient\":[1,1],\"constraint_rhs\":3}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([0, 0], [1, 1], 3, 3/2, [3/2, 3/2], 3, [3, 3], 是, 是, squared_distance_with_one_regular_linear_constraint_solved_exactly)","explanation_zh":"Wave6第十三層核對=([0, 0], [1, 1], 3, 3/2, [3/2, 3/2], 3, [3, 3], 是, 是, squared_distance_with_one_regular_linear_constraint_solved_exactly)；依完整明示資料精確重算；regular equality-constrained local extremum只得到必要multiplier condition；stationarity非充分；rank deficient是不適用而非無極值，且本條件不是inequality KKT、global或uniqueness證書。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Lagrange乘子條件"]},"variant":{"family_id":"qf-mod50-lagrange-single-linear-constraint","seed":202611194051,"parameters":{"center":[0,0],"constraint_gradient":[1,1],"constraint_rhs":3,"domain":"Exact_equality_constraint_Lagrange_condition","operation":"lagrange-single-linear-constraint","semantic_template":"lagrange-single-linear-constraint|{\"center\":[[0,1],[0,1]],\"constraint_gradient\":[[1,1],[1,1]],\"constraint_rhs\":[3,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r04-0052","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-multiplier-condition"],"title_zh":"Lagrange乘子條件","stem_zh":"feasibility、constraint-gradient full row rank與grad_f=J_g^T*lambda convention。操作=lagrange-single-linear-constraint；明示資料={\"center\":[1,1],\"constraint_gradient\":[1,2],\"constraint_rhs\":4}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([1, 1], [1, 2], 4, 1/5, [6/5, 7/5], 2/5, [2/5, 4/5], 是, 是, squared_distance_with_one_regular_linear_constraint_solved_exactly)","explanation_zh":"Wave6第十三層核對=([1, 1], [1, 2], 4, 1/5, [6/5, 7/5], 2/5, [2/5, 4/5], 是, 是, squared_distance_with_one_regular_linear_constraint_solved_exactly)；依完整明示資料精確重算；regular equality-constrained local extremum只得到必要multiplier condition；stationarity非充分；rank deficient是不適用而非無極值，且本條件不是inequality KKT、global或uniqueness證書。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Lagrange乘子條件"]},"variant":{"family_id":"qf-mod50-lagrange-single-linear-constraint","seed":202611194052,"parameters":{"center":[1,1],"constraint_gradient":[1,2],"constraint_rhs":4,"domain":"Exact_equality_constraint_Lagrange_condition","operation":"lagrange-single-linear-constraint","semantic_template":"lagrange-single-linear-constraint|{\"center\":[[1,1],[1,1]],\"constraint_gradient\":[[1,1],[2,1]],\"constraint_rhs\":[4,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r04-0053","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-multiplier-condition"],"title_zh":"Lagrange乘子條件","stem_zh":"feasibility、constraint-gradient full row rank與grad_f=J_g^T*lambda convention。操作=lagrange-single-linear-constraint；明示資料={\"center\":[2,2],\"constraint_gradient\":[1,3],\"constraint_rhs\":5}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([2, 2], [1, 3], 5, -3/10, [17/10, 11/10], -3/5, [-3/5, -9/5], 是, 是, squared_distance_with_one_regular_linear_constraint_solved_exactly)","explanation_zh":"Wave6第十三層核對=([2, 2], [1, 3], 5, -3/10, [17/10, 11/10], -3/5, [-3/5, -9/5], 是, 是, squared_distance_with_one_regular_linear_constraint_solved_exactly)；依完整明示資料精確重算；regular equality-constrained local extremum只得到必要multiplier condition；stationarity非充分；rank deficient是不適用而非無極值，且本條件不是inequality KKT、global或uniqueness證書。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Lagrange乘子條件"]},"variant":{"family_id":"qf-mod50-lagrange-single-linear-constraint","seed":202611194053,"parameters":{"center":[2,2],"constraint_gradient":[1,3],"constraint_rhs":5,"domain":"Exact_equality_constraint_Lagrange_condition","operation":"lagrange-single-linear-constraint","semantic_template":"lagrange-single-linear-constraint|{\"center\":[[2,1],[2,1]],\"constraint_gradient\":[[1,1],[3,1]],\"constraint_rhs\":[5,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r04-0054","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-multiplier-condition"],"title_zh":"Lagrange乘子條件","stem_zh":"feasibility、constraint-gradient full row rank與grad_f=J_g^T*lambda convention。操作=lagrange-single-linear-constraint；明示資料={\"center\":[3,3],\"constraint_gradient\":[1,4],\"constraint_rhs\":6}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([3, 3], [1, 4], 6, -9/17, [42/17, 15/17], -18/17, [-18/17, -72/17], 是, 是, squared_distance_with_one_regular_linear_constraint_solved_exactly)","explanation_zh":"Wave6第十三層核對=([3, 3], [1, 4], 6, -9/17, [42/17, 15/17], -18/17, [-18/17, -72/17], 是, 是, squared_distance_with_one_regular_linear_constraint_solved_exactly)；依完整明示資料精確重算；regular equality-constrained local extremum只得到必要multiplier condition；stationarity非充分；rank deficient是不適用而非無極值，且本條件不是inequality KKT、global或uniqueness證書。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Lagrange乘子條件"]},"variant":{"family_id":"qf-mod50-lagrange-single-linear-constraint","seed":202611194054,"parameters":{"center":[3,3],"constraint_gradient":[1,4],"constraint_rhs":6,"domain":"Exact_equality_constraint_Lagrange_condition","operation":"lagrange-single-linear-constraint","semantic_template":"lagrange-single-linear-constraint|{\"center\":[[3,1],[3,1]],\"constraint_gradient\":[[1,1],[4,1]],\"constraint_rhs\":[6,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r04-0055","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-multiplier-condition"],"title_zh":"Lagrange乘子條件","stem_zh":"feasibility、constraint-gradient full row rank與grad_f=J_g^T*lambda convention。操作=lagrange-single-linear-constraint；明示資料={\"center\":[4,4],\"constraint_gradient\":[1,5],\"constraint_rhs\":7}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([4, 4], [1, 5], 7, -17/26, [87/26, 19/26], -17/13, [-17/13, -85/13], 是, 是, squared_distance_with_one_regular_linear_constraint_solved_exactly)","explanation_zh":"Wave6第十三層核對=([4, 4], [1, 5], 7, -17/26, [87/26, 19/26], -17/13, [-17/13, -85/13], 是, 是, squared_distance_with_one_regular_linear_constraint_solved_exactly)；依完整明示資料精確重算；regular equality-constrained local extremum只得到必要multiplier condition；stationarity非充分；rank deficient是不適用而非無極值，且本條件不是inequality KKT、global或uniqueness證書。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Lagrange乘子條件"]},"variant":{"family_id":"qf-mod50-lagrange-single-linear-constraint","seed":202611194055,"parameters":{"center":[4,4],"constraint_gradient":[1,5],"constraint_rhs":7,"domain":"Exact_equality_constraint_Lagrange_condition","operation":"lagrange-single-linear-constraint","semantic_template":"lagrange-single-linear-constraint|{\"center\":[[4,1],[4,1]],\"constraint_gradient\":[[1,1],[5,1]],\"constraint_rhs\":[7,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r04-0056","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-multiplier-condition"],"title_zh":"Lagrange乘子條件","stem_zh":"feasibility、constraint-gradient full row rank與grad_f=J_g^T*lambda convention。操作=lagrange-single-linear-constraint；明示資料={\"center\":[0,5],\"constraint_gradient\":[1,6],\"constraint_rhs\":8}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([0, 5], [1, 6], 8, -22/37, [-22/37, 53/37], -44/37, [-44/37, -264/37], 是, 是, squared_distance_with_one_regular_linear_constraint_solved_exactly)","explanation_zh":"Wave6第十三層核對=([0, 5], [1, 6], 8, -22/37, [-22/37, 53/37], -44/37, [-44/37, -264/37], 是, 是, squared_distance_with_one_regular_linear_constraint_solved_exactly)；依完整明示資料精確重算；regular equality-constrained local extremum只得到必要multiplier condition；stationarity非充分；rank deficient是不適用而非無極值，且本條件不是inequality KKT、global或uniqueness證書。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Lagrange乘子條件"]},"variant":{"family_id":"qf-mod50-lagrange-single-linear-constraint","seed":202611194056,"parameters":{"center":[0,5],"constraint_gradient":[1,6],"constraint_rhs":8,"domain":"Exact_equality_constraint_Lagrange_condition","operation":"lagrange-single-linear-constraint","semantic_template":"lagrange-single-linear-constraint|{\"center\":[[0,1],[5,1]],\"constraint_gradient\":[[1,1],[6,1]],\"constraint_rhs\":[8,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r04-0057","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-multiplier-condition"],"title_zh":"Lagrange乘子條件","stem_zh":"feasibility、constraint-gradient full row rank與grad_f=J_g^T*lambda convention。操作=lagrange-single-linear-constraint；明示資料={\"center\":[1,6],\"constraint_gradient\":[1,7],\"constraint_rhs\":9}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([1, 6], [1, 7], 9, -17/25, [8/25, 31/25], -34/25, [-34/25, -238/25], 是, 是, squared_distance_with_one_regular_linear_constraint_solved_exactly)","explanation_zh":"Wave6第十三層核對=([1, 6], [1, 7], 9, -17/25, [8/25, 31/25], -34/25, [-34/25, -238/25], 是, 是, squared_distance_with_one_regular_linear_constraint_solved_exactly)；依完整明示資料精確重算；regular equality-constrained local extremum只得到必要multiplier condition；stationarity非充分；rank deficient是不適用而非無極值，且本條件不是inequality KKT、global或uniqueness證書。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Lagrange乘子條件"]},"variant":{"family_id":"qf-mod50-lagrange-single-linear-constraint","seed":202611194057,"parameters":{"center":[1,6],"constraint_gradient":[1,7],"constraint_rhs":9,"domain":"Exact_equality_constraint_Lagrange_condition","operation":"lagrange-single-linear-constraint","semantic_template":"lagrange-single-linear-constraint|{\"center\":[[1,1],[6,1]],\"constraint_gradient\":[[1,1],[7,1]],\"constraint_rhs\":[9,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r04-0058","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-multiplier-condition"],"title_zh":"Lagrange乘子條件","stem_zh":"feasibility、constraint-gradient full row rank與grad_f=J_g^T*lambda convention。操作=lagrange-single-linear-constraint；明示資料={\"center\":[2,0],\"constraint_gradient\":[1,8],\"constraint_rhs\":10}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([2, 0], [1, 8], 10, 8/65, [138/65, 64/65], 16/65, [16/65, 128/65], 是, 是, squared_distance_with_one_regular_linear_constraint_solved_exactly)","explanation_zh":"Wave6第十三層核對=([2, 0], [1, 8], 10, 8/65, [138/65, 64/65], 16/65, [16/65, 128/65], 是, 是, squared_distance_with_one_regular_linear_constraint_solved_exactly)；依完整明示資料精確重算；regular equality-constrained local extremum只得到必要multiplier condition；stationarity非充分；rank deficient是不適用而非無極值，且本條件不是inequality KKT、global或uniqueness證書。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Lagrange乘子條件"]},"variant":{"family_id":"qf-mod50-lagrange-single-linear-constraint","seed":202611194058,"parameters":{"center":[2,0],"constraint_gradient":[1,8],"constraint_rhs":10,"domain":"Exact_equality_constraint_Lagrange_condition","operation":"lagrange-single-linear-constraint","semantic_template":"lagrange-single-linear-constraint|{\"center\":[[2,1],[0,1]],\"constraint_gradient\":[[1,1],[8,1]],\"constraint_rhs\":[10,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r04-0059","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-multiplier-condition"],"title_zh":"Lagrange乘子條件","stem_zh":"feasibility、constraint-gradient full row rank與grad_f=J_g^T*lambda convention。操作=lagrange-single-linear-constraint；明示資料={\"center\":[3,1],\"constraint_gradient\":[1,9],\"constraint_rhs\":11}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([3, 1], [1, 9], 11, -1/82, [245/82, 73/82], -1/41, [-1/41, -9/41], 是, 是, squared_distance_with_one_regular_linear_constraint_solved_exactly)","explanation_zh":"Wave6第十三層核對=([3, 1], [1, 9], 11, -1/82, [245/82, 73/82], -1/41, [-1/41, -9/41], 是, 是, squared_distance_with_one_regular_linear_constraint_solved_exactly)；依完整明示資料精確重算；regular equality-constrained local extremum只得到必要multiplier condition；stationarity非充分；rank deficient是不適用而非無極值，且本條件不是inequality KKT、global或uniqueness證書。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Lagrange乘子條件"]},"variant":{"family_id":"qf-mod50-lagrange-single-linear-constraint","seed":202611194059,"parameters":{"center":[3,1],"constraint_gradient":[1,9],"constraint_rhs":11,"domain":"Exact_equality_constraint_Lagrange_condition","operation":"lagrange-single-linear-constraint","semantic_template":"lagrange-single-linear-constraint|{\"center\":[[3,1],[1,1]],\"constraint_gradient\":[[1,1],[9,1]],\"constraint_rhs\":[11,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r04-0060","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-multiplier-condition"],"title_zh":"Lagrange乘子條件","stem_zh":"feasibility、constraint-gradient full row rank與grad_f=J_g^T*lambda convention。操作=lagrange-single-linear-constraint；明示資料={\"center\":[4,2],\"constraint_gradient\":[1,10],\"constraint_rhs\":12}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([4, 2], [1, 10], 12, -12/101, [392/101, 82/101], -24/101, [-24/101, -240/101], 是, 是, squared_distance_with_one_regular_linear_constraint_solved_exactly)","explanation_zh":"Wave6第十三層核對=([4, 2], [1, 10], 12, -12/101, [392/101, 82/101], -24/101, [-24/101, -240/101], 是, 是, squared_distance_with_one_regular_linear_constraint_solved_exactly)；依完整明示資料精確重算；regular equality-constrained local extremum只得到必要multiplier condition；stationarity非充分；rank deficient是不適用而非無極值，且本條件不是inequality KKT、global或uniqueness證書。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Lagrange乘子條件"]},"variant":{"family_id":"qf-mod50-lagrange-single-linear-constraint","seed":202611194060,"parameters":{"center":[4,2],"constraint_gradient":[1,10],"constraint_rhs":12,"domain":"Exact_equality_constraint_Lagrange_condition","operation":"lagrange-single-linear-constraint","semantic_template":"lagrange-single-linear-constraint|{\"center\":[[4,1],[2,1]],\"constraint_gradient\":[[1,1],[10,1]],\"constraint_rhs\":[12,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r04-0061","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-multiplier-condition"],"title_zh":"Lagrange乘子條件","stem_zh":"feasibility、constraint-gradient full row rank與grad_f=J_g^T*lambda convention。操作=lagrange-single-linear-constraint；明示資料={\"center\":[0,3],\"constraint_gradient\":[1,11],\"constraint_rhs\":13}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([0, 3], [1, 11], 13, -10/61, [-10/61, 73/61], -20/61, [-20/61, -220/61], 是, 是, squared_distance_with_one_regular_linear_constraint_solved_exactly)","explanation_zh":"Wave6第十三層核對=([0, 3], [1, 11], 13, -10/61, [-10/61, 73/61], -20/61, [-20/61, -220/61], 是, 是, squared_distance_with_one_regular_linear_constraint_solved_exactly)；依完整明示資料精確重算；regular equality-constrained local extremum只得到必要multiplier condition；stationarity非充分；rank deficient是不適用而非無極值，且本條件不是inequality KKT、global或uniqueness證書。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Lagrange乘子條件"]},"variant":{"family_id":"qf-mod50-lagrange-single-linear-constraint","seed":202611194061,"parameters":{"center":[0,3],"constraint_gradient":[1,11],"constraint_rhs":13,"domain":"Exact_equality_constraint_Lagrange_condition","operation":"lagrange-single-linear-constraint","semantic_template":"lagrange-single-linear-constraint|{\"center\":[[0,1],[3,1]],\"constraint_gradient\":[[1,1],[11,1]],\"constraint_rhs\":[13,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r04-0062","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-multiplier-condition"],"title_zh":"Lagrange乘子條件","stem_zh":"feasibility、constraint-gradient full row rank與grad_f=J_g^T*lambda convention。操作=lagrange-single-linear-constraint；明示資料={\"center\":[1,4],\"constraint_gradient\":[1,12],\"constraint_rhs\":14}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([1, 4], [1, 12], 14, -7/29, [22/29, 32/29], -14/29, [-14/29, -168/29], 是, 是, squared_distance_with_one_regular_linear_constraint_solved_exactly)","explanation_zh":"Wave6第十三層核對=([1, 4], [1, 12], 14, -7/29, [22/29, 32/29], -14/29, [-14/29, -168/29], 是, 是, squared_distance_with_one_regular_linear_constraint_solved_exactly)；依完整明示資料精確重算；regular equality-constrained local extremum只得到必要multiplier condition；stationarity非充分；rank deficient是不適用而非無極值，且本條件不是inequality KKT、global或uniqueness證書。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Lagrange乘子條件"]},"variant":{"family_id":"qf-mod50-lagrange-single-linear-constraint","seed":202611194062,"parameters":{"center":[1,4],"constraint_gradient":[1,12],"constraint_rhs":14,"domain":"Exact_equality_constraint_Lagrange_condition","operation":"lagrange-single-linear-constraint","semantic_template":"lagrange-single-linear-constraint|{\"center\":[[1,1],[4,1]],\"constraint_gradient\":[[1,1],[12,1]],\"constraint_rhs\":[14,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r04-0063","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-multiplier-condition"],"title_zh":"Lagrange乘子條件","stem_zh":"feasibility、constraint-gradient full row rank與grad_f=J_g^T*lambda convention。操作=lagrange-single-linear-constraint；明示資料={\"center\":[2,5],\"constraint_gradient\":[1,13],\"constraint_rhs\":15}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([2, 5], [1, 13], 15, -26/85, [144/85, 87/85], -52/85, [-52/85, -676/85], 是, 是, squared_distance_with_one_regular_linear_constraint_solved_exactly)","explanation_zh":"Wave6第十三層核對=([2, 5], [1, 13], 15, -26/85, [144/85, 87/85], -52/85, [-52/85, -676/85], 是, 是, squared_distance_with_one_regular_linear_constraint_solved_exactly)；依完整明示資料精確重算；regular equality-constrained local extremum只得到必要multiplier condition；stationarity非充分；rank deficient是不適用而非無極值，且本條件不是inequality KKT、global或uniqueness證書。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Lagrange乘子條件"]},"variant":{"family_id":"qf-mod50-lagrange-single-linear-constraint","seed":202611194063,"parameters":{"center":[2,5],"constraint_gradient":[1,13],"constraint_rhs":15,"domain":"Exact_equality_constraint_Lagrange_condition","operation":"lagrange-single-linear-constraint","semantic_template":"lagrange-single-linear-constraint|{\"center\":[[2,1],[5,1]],\"constraint_gradient\":[[1,1],[13,1]],\"constraint_rhs\":[15,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r04-0064","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-multiplier-condition"],"title_zh":"Lagrange乘子條件","stem_zh":"feasibility、constraint-gradient full row rank與grad_f=J_g^T*lambda convention。操作=lagrange-single-linear-constraint；明示資料={\"center\":[3,6],\"constraint_gradient\":[1,14],\"constraint_rhs\":16}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([3, 6], [1, 14], 16, -71/197, [520/197, 188/197], -142/197, [-142/197, -1988/197], 是, 是, squared_distance_with_one_regular_linear_constraint_solved_exactly)","explanation_zh":"Wave6第十三層核對=([3, 6], [1, 14], 16, -71/197, [520/197, 188/197], -142/197, [-142/197, -1988/197], 是, 是, squared_distance_with_one_regular_linear_constraint_solved_exactly)；依完整明示資料精確重算；regular equality-constrained local extremum只得到必要multiplier condition；stationarity非充分；rank deficient是不適用而非無極值，且本條件不是inequality KKT、global或uniqueness證書。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Lagrange乘子條件"]},"variant":{"family_id":"qf-mod50-lagrange-single-linear-constraint","seed":202611194064,"parameters":{"center":[3,6],"constraint_gradient":[1,14],"constraint_rhs":16,"domain":"Exact_equality_constraint_Lagrange_condition","operation":"lagrange-single-linear-constraint","semantic_template":"lagrange-single-linear-constraint|{\"center\":[[3,1],[6,1]],\"constraint_gradient\":[[1,1],[14,1]],\"constraint_rhs\":[16,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r04-0065","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-multiplier-condition"],"title_zh":"Lagrange乘子條件","stem_zh":"feasibility、constraint-gradient full row rank與grad_f=J_g^T*lambda convention。操作=lagrange-single-linear-constraint；明示資料={\"center\":[4,0],\"constraint_gradient\":[1,15],\"constraint_rhs\":17}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([4, 0], [1, 15], 17, 13/226, [917/226, 195/226], 13/113, [13/113, 195/113], 是, 是, squared_distance_with_one_regular_linear_constraint_solved_exactly)","explanation_zh":"Wave6第十三層核對=([4, 0], [1, 15], 17, 13/226, [917/226, 195/226], 13/113, [13/113, 195/113], 是, 是, squared_distance_with_one_regular_linear_constraint_solved_exactly)；依完整明示資料精確重算；regular equality-constrained local extremum只得到必要multiplier condition；stationarity非充分；rank deficient是不適用而非無極值，且本條件不是inequality KKT、global或uniqueness證書。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Lagrange乘子條件"]},"variant":{"family_id":"qf-mod50-lagrange-single-linear-constraint","seed":202611194065,"parameters":{"center":[4,0],"constraint_gradient":[1,15],"constraint_rhs":17,"domain":"Exact_equality_constraint_Lagrange_condition","operation":"lagrange-single-linear-constraint","semantic_template":"lagrange-single-linear-constraint|{\"center\":[[4,1],[0,1]],\"constraint_gradient\":[[1,1],[15,1]],\"constraint_rhs\":[17,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r04-0066","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-multiplier-condition"],"title_zh":"Lagrange乘子條件","stem_zh":"feasibility、constraint-gradient full row rank與grad_f=J_g^T*lambda convention。操作=lagrange-single-linear-constraint；明示資料={\"center\":[0,1],\"constraint_gradient\":[1,16],\"constraint_rhs\":18}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([0, 1], [1, 16], 18, 2/257, [2/257, 289/257], 4/257, [4/257, 64/257], 是, 是, squared_distance_with_one_regular_linear_constraint_solved_exactly)","explanation_zh":"Wave6第十三層核對=([0, 1], [1, 16], 18, 2/257, [2/257, 289/257], 4/257, [4/257, 64/257], 是, 是, squared_distance_with_one_regular_linear_constraint_solved_exactly)；依完整明示資料精確重算；regular equality-constrained local extremum只得到必要multiplier condition；stationarity非充分；rank deficient是不適用而非無極值，且本條件不是inequality KKT、global或uniqueness證書。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Lagrange乘子條件"]},"variant":{"family_id":"qf-mod50-lagrange-single-linear-constraint","seed":202611194066,"parameters":{"center":[0,1],"constraint_gradient":[1,16],"constraint_rhs":18,"domain":"Exact_equality_constraint_Lagrange_condition","operation":"lagrange-single-linear-constraint","semantic_template":"lagrange-single-linear-constraint|{\"center\":[[0,1],[1,1]],\"constraint_gradient\":[[1,1],[16,1]],\"constraint_rhs\":[18,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r04-0067","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-multiplier-condition"],"title_zh":"Lagrange乘子條件","stem_zh":"feasibility、constraint-gradient full row rank與grad_f=J_g^T*lambda convention。操作=lagrange-single-linear-constraint；明示資料={\"center\":[1,2],\"constraint_gradient\":[1,17],\"constraint_rhs\":19}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([1, 2], [1, 17], 19, -8/145, [137/145, 154/145], -16/145, [-16/145, -272/145], 是, 是, squared_distance_with_one_regular_linear_constraint_solved_exactly)","explanation_zh":"Wave6第十三層核對=([1, 2], [1, 17], 19, -8/145, [137/145, 154/145], -16/145, [-16/145, -272/145], 是, 是, squared_distance_with_one_regular_linear_constraint_solved_exactly)；依完整明示資料精確重算；regular equality-constrained local extremum只得到必要multiplier condition；stationarity非充分；rank deficient是不適用而非無極值，且本條件不是inequality KKT、global或uniqueness證書。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Lagrange乘子條件"]},"variant":{"family_id":"qf-mod50-lagrange-single-linear-constraint","seed":202611194067,"parameters":{"center":[1,2],"constraint_gradient":[1,17],"constraint_rhs":19,"domain":"Exact_equality_constraint_Lagrange_condition","operation":"lagrange-single-linear-constraint","semantic_template":"lagrange-single-linear-constraint|{\"center\":[[1,1],[2,1]],\"constraint_gradient\":[[1,1],[17,1]],\"constraint_rhs\":[19,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r04-0068","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-multiplier-condition"],"title_zh":"Lagrange乘子條件","stem_zh":"feasibility、constraint-gradient full row rank與grad_f=J_g^T*lambda convention。操作=lagrange-single-linear-constraint；明示資料={\"center\":[2,3],\"constraint_gradient\":[1,18],\"constraint_rhs\":20}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([2, 3], [1, 18], 20, -36/325, [614/325, 327/325], -72/325, [-72/325, -1296/325], 是, 是, squared_distance_with_one_regular_linear_constraint_solved_exactly)","explanation_zh":"Wave6第十三層核對=([2, 3], [1, 18], 20, -36/325, [614/325, 327/325], -72/325, [-72/325, -1296/325], 是, 是, squared_distance_with_one_regular_linear_constraint_solved_exactly)；依完整明示資料精確重算；regular equality-constrained local extremum只得到必要multiplier condition；stationarity非充分；rank deficient是不適用而非無極值，且本條件不是inequality KKT、global或uniqueness證書。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Lagrange乘子條件"]},"variant":{"family_id":"qf-mod50-lagrange-single-linear-constraint","seed":202611194068,"parameters":{"center":[2,3],"constraint_gradient":[1,18],"constraint_rhs":20,"domain":"Exact_equality_constraint_Lagrange_condition","operation":"lagrange-single-linear-constraint","semantic_template":"lagrange-single-linear-constraint|{\"center\":[[2,1],[3,1]],\"constraint_gradient\":[[1,1],[18,1]],\"constraint_rhs\":[20,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r04-0069","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-multiplier-condition"],"title_zh":"Lagrange乘子條件","stem_zh":"feasibility、constraint-gradient full row rank與grad_f=J_g^T*lambda convention。操作=lagrange-single-linear-constraint；明示資料={\"center\":[3,4],\"constraint_gradient\":[1,19],\"constraint_rhs\":21}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([3, 4], [1, 19], 21, -29/181, [514/181, 173/181], -58/181, [-58/181, -1102/181], 是, 是, squared_distance_with_one_regular_linear_constraint_solved_exactly)","explanation_zh":"Wave6第十三層核對=([3, 4], [1, 19], 21, -29/181, [514/181, 173/181], -58/181, [-58/181, -1102/181], 是, 是, squared_distance_with_one_regular_linear_constraint_solved_exactly)；依完整明示資料精確重算；regular equality-constrained local extremum只得到必要multiplier condition；stationarity非充分；rank deficient是不適用而非無極值，且本條件不是inequality KKT、global或uniqueness證書。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Lagrange乘子條件"]},"variant":{"family_id":"qf-mod50-lagrange-single-linear-constraint","seed":202611194069,"parameters":{"center":[3,4],"constraint_gradient":[1,19],"constraint_rhs":21,"domain":"Exact_equality_constraint_Lagrange_condition","operation":"lagrange-single-linear-constraint","semantic_template":"lagrange-single-linear-constraint|{\"center\":[[3,1],[4,1]],\"constraint_gradient\":[[1,1],[19,1]],\"constraint_rhs\":[21,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r04-0070","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-multiplier-condition"],"title_zh":"Lagrange乘子條件","stem_zh":"feasibility、constraint-gradient full row rank與grad_f=J_g^T*lambda convention。操作=lagrange-single-linear-constraint；明示資料={\"center\":[4,5],\"constraint_gradient\":[1,20],\"constraint_rhs\":22}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([4, 5], [1, 20], 22, -82/401, [1522/401, 365/401], -164/401, [-164/401, -3280/401], 是, 是, squared_distance_with_one_regular_linear_constraint_solved_exactly)","explanation_zh":"Wave6第十三層核對=([4, 5], [1, 20], 22, -82/401, [1522/401, 365/401], -164/401, [-164/401, -3280/401], 是, 是, squared_distance_with_one_regular_linear_constraint_solved_exactly)；依完整明示資料精確重算；regular equality-constrained local extremum只得到必要multiplier condition；stationarity非充分；rank deficient是不適用而非無極值，且本條件不是inequality KKT、global或uniqueness證書。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Lagrange乘子條件"]},"variant":{"family_id":"qf-mod50-lagrange-single-linear-constraint","seed":202611194070,"parameters":{"center":[4,5],"constraint_gradient":[1,20],"constraint_rhs":22,"domain":"Exact_equality_constraint_Lagrange_condition","operation":"lagrange-single-linear-constraint","semantic_template":"lagrange-single-linear-constraint|{\"center\":[[4,1],[5,1]],\"constraint_gradient\":[[1,1],[20,1]],\"constraint_rhs\":[22,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r04-0071","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-multiplier-condition"],"title_zh":"Lagrange乘子條件","stem_zh":"feasibility、constraint-gradient full row rank與grad_f=J_g^T*lambda convention。操作=lagrange-single-linear-constraint；明示資料={\"center\":[0,6],\"constraint_gradient\":[1,21],\"constraint_rhs\":23}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([0, 6], [1, 21], 23, -103/442, [-103/442, 489/442], -103/221, [-103/221, -2163/221], 是, 是, squared_distance_with_one_regular_linear_constraint_solved_exactly)","explanation_zh":"Wave6第十三層核對=([0, 6], [1, 21], 23, -103/442, [-103/442, 489/442], -103/221, [-103/221, -2163/221], 是, 是, squared_distance_with_one_regular_linear_constraint_solved_exactly)；依完整明示資料精確重算；regular equality-constrained local extremum只得到必要multiplier condition；stationarity非充分；rank deficient是不適用而非無極值，且本條件不是inequality KKT、global或uniqueness證書。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Lagrange乘子條件"]},"variant":{"family_id":"qf-mod50-lagrange-single-linear-constraint","seed":202611194071,"parameters":{"center":[0,6],"constraint_gradient":[1,21],"constraint_rhs":23,"domain":"Exact_equality_constraint_Lagrange_condition","operation":"lagrange-single-linear-constraint","semantic_template":"lagrange-single-linear-constraint|{\"center\":[[0,1],[6,1]],\"constraint_gradient\":[[1,1],[21,1]],\"constraint_rhs\":[23,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r04-0072","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-multiplier-condition"],"title_zh":"Lagrange乘子條件","stem_zh":"feasibility、constraint-gradient full row rank與grad_f=J_g^T*lambda convention。操作=lagrange-single-linear-constraint；明示資料={\"center\":[1,0],\"constraint_gradient\":[1,22],\"constraint_rhs\":24}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([1, 0], [1, 22], 24, 23/485, [508/485, 506/485], 46/485, [46/485, 1012/485], 是, 是, squared_distance_with_one_regular_linear_constraint_solved_exactly)","explanation_zh":"Wave6第十三層核對=([1, 0], [1, 22], 24, 23/485, [508/485, 506/485], 46/485, [46/485, 1012/485], 是, 是, squared_distance_with_one_regular_linear_constraint_solved_exactly)；依完整明示資料精確重算；regular equality-constrained local extremum只得到必要multiplier condition；stationarity非充分；rank deficient是不適用而非無極值，且本條件不是inequality KKT、global或uniqueness證書。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Lagrange乘子條件"]},"variant":{"family_id":"qf-mod50-lagrange-single-linear-constraint","seed":202611194072,"parameters":{"center":[1,0],"constraint_gradient":[1,22],"constraint_rhs":24,"domain":"Exact_equality_constraint_Lagrange_condition","operation":"lagrange-single-linear-constraint","semantic_template":"lagrange-single-linear-constraint|{\"center\":[[1,1],[0,1]],\"constraint_gradient\":[[1,1],[22,1]],\"constraint_rhs\":[24,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r04-0073","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-multiplier-condition"],"title_zh":"Lagrange乘子條件","stem_zh":"feasibility、constraint-gradient full row rank與grad_f=J_g^T*lambda convention。操作=lagrange-single-linear-constraint；明示資料={\"center\":[2,1],\"constraint_gradient\":[1,23],\"constraint_rhs\":25}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([2, 1], [1, 23], 25, 0, [2, 1], 0, [0, 0], 是, 是, squared_distance_with_one_regular_linear_constraint_solved_exactly)","explanation_zh":"Wave6第十三層核對=([2, 1], [1, 23], 25, 0, [2, 1], 0, [0, 0], 是, 是, squared_distance_with_one_regular_linear_constraint_solved_exactly)；依完整明示資料精確重算；regular equality-constrained local extremum只得到必要multiplier condition；stationarity非充分；rank deficient是不適用而非無極值，且本條件不是inequality KKT、global或uniqueness證書。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Lagrange乘子條件"]},"variant":{"family_id":"qf-mod50-lagrange-single-linear-constraint","seed":202611194073,"parameters":{"center":[2,1],"constraint_gradient":[1,23],"constraint_rhs":25,"domain":"Exact_equality_constraint_Lagrange_condition","operation":"lagrange-single-linear-constraint","semantic_template":"lagrange-single-linear-constraint|{\"center\":[[2,1],[1,1]],\"constraint_gradient\":[[1,1],[23,1]],\"constraint_rhs\":[25,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r04-0074","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-multiplier-condition"],"title_zh":"Lagrange乘子條件","stem_zh":"feasibility、constraint-gradient full row rank與grad_f=J_g^T*lambda convention。操作=lagrange-single-linear-constraint；明示資料={\"center\":[3,2],\"constraint_gradient\":[1,24],\"constraint_rhs\":26}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([3, 2], [1, 24], 26, -25/577, [1706/577, 554/577], -50/577, [-50/577, -1200/577], 是, 是, squared_distance_with_one_regular_linear_constraint_solved_exactly)","explanation_zh":"Wave6第十三層核對=([3, 2], [1, 24], 26, -25/577, [1706/577, 554/577], -50/577, [-50/577, -1200/577], 是, 是, squared_distance_with_one_regular_linear_constraint_solved_exactly)；依完整明示資料精確重算；regular equality-constrained local extremum只得到必要multiplier condition；stationarity非充分；rank deficient是不適用而非無極值，且本條件不是inequality KKT、global或uniqueness證書。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Lagrange乘子條件"]},"variant":{"family_id":"qf-mod50-lagrange-single-linear-constraint","seed":202611194074,"parameters":{"center":[3,2],"constraint_gradient":[1,24],"constraint_rhs":26,"domain":"Exact_equality_constraint_Lagrange_condition","operation":"lagrange-single-linear-constraint","semantic_template":"lagrange-single-linear-constraint|{\"center\":[[3,1],[2,1]],\"constraint_gradient\":[[1,1],[24,1]],\"constraint_rhs\":[26,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r04-0075","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-multiplier-condition"],"title_zh":"Lagrange乘子條件","stem_zh":"feasibility、constraint-gradient full row rank與grad_f=J_g^T*lambda convention。操作=lagrange-single-linear-constraint；明示資料={\"center\":[4,3],\"constraint_gradient\":[1,25],\"constraint_rhs\":27}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"([4, 3], [1, 25], 27, -26/313, [1226/313, 289/313], -52/313, [-52/313, -1300/313], 是, 是, squared_distance_with_one_regular_linear_constraint_solved_exactly)","explanation_zh":"Wave6第十三層核對=([4, 3], [1, 25], 27, -26/313, [1226/313, 289/313], -52/313, [-52/313, -1300/313], 是, 是, squared_distance_with_one_regular_linear_constraint_solved_exactly)；依完整明示資料精確重算；regular equality-constrained local extremum只得到必要multiplier condition；stationarity非充分；rank deficient是不適用而非無極值，且本條件不是inequality KKT、global或uniqueness證書。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Lagrange乘子條件"]},"variant":{"family_id":"qf-mod50-lagrange-single-linear-constraint","seed":202611194075,"parameters":{"center":[4,3],"constraint_gradient":[1,25],"constraint_rhs":27,"domain":"Exact_equality_constraint_Lagrange_condition","operation":"lagrange-single-linear-constraint","semantic_template":"lagrange-single-linear-constraint|{\"center\":[[4,1],[3,1]],\"constraint_gradient\":[[1,1],[25,1]],\"constraint_rhs\":[27,1]}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r04-0076","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-multiplier-condition"],"title_zh":"Lagrange乘子條件","stem_zh":"feasibility、constraint-gradient full row rank與grad_f=J_g^T*lambda convention。操作=lagrange-boundary；明示資料={\"mode\":\"rank_deficient\",\"model\":{\"constraint_gradients\":[[1,1],[2,2]]}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(rank_deficient, [[1, 1], [2, 2]], 1, 2, 否, regular_Lagrange_theorem_not_applicable_does_not_mean_no_extremum)","explanation_zh":"Wave6第十三層核對=(rank_deficient, [[1, 1], [2, 2]], 1, 2, 否, regular_Lagrange_theorem_not_applicable_does_not_mean_no_extremum)；依完整明示資料精確重算；regular equality-constrained local extremum只得到必要multiplier condition；stationarity非充分；rank deficient是不適用而非無極值，且本條件不是inequality KKT、global或uniqueness證書。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Lagrange乘子條件"]},"variant":{"family_id":"qf-mod50-lagrange-boundary","seed":202611194076,"parameters":{"mode":"rank_deficient","model":{"constraint_gradients":[[1,1],[2,2]]},"domain":"Exact_equality_constraint_Lagrange_condition","operation":"lagrange-boundary","semantic_template":"lagrange-boundary|{\"mode\":\"rank_deficient\",\"model\":{\"constraint_gradients\":[[[1,1],[1,1]],[[2,1],[2,1]]]}}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r04-0077","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-multiplier-condition"],"title_zh":"Lagrange乘子條件","stem_zh":"feasibility、constraint-gradient full row rank與grad_f=J_g^T*lambda convention。操作=lagrange-boundary；明示資料={\"mode\":\"stationary_not_sufficient\",\"model\":{\"scale\":2}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(stationary_not_sufficient, f(x,y)=-2*x^2, g(x,y)=y=0, [0,0], grad_f=[0,0]=0*grad_g, 是, strict_local_maximum_on_constraint, 否, first_order_multiplier_equation_is_necessary_not_sufficient_for_a_minimum)","explanation_zh":"Wave6第十三層核對=(stationary_not_sufficient, f(x,y)=-2*x^2, g(x,y)=y=0, [0,0], grad_f=[0,0]=0*grad_g, 是, strict_local_maximum_on_constraint, 否, first_order_multiplier_equation_is_necessary_not_sufficient_for_a_minimum)；依完整明示資料精確重算；regular equality-constrained local extremum只得到必要multiplier condition；stationarity非充分；rank deficient是不適用而非無極值，且本條件不是inequality KKT、global或uniqueness證書。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Lagrange乘子條件"]},"variant":{"family_id":"qf-mod50-lagrange-boundary","seed":202611194077,"parameters":{"mode":"stationary_not_sufficient","model":{"scale":2},"domain":"Exact_equality_constraint_Lagrange_condition","operation":"lagrange-boundary","semantic_template":"lagrange-boundary|{\"mode\":\"stationary_not_sufficient\",\"model\":{\"scale\":[2,1]}}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r04-0078","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-multiplier-condition"],"title_zh":"Lagrange乘子條件","stem_zh":"feasibility、constraint-gradient full row rank與grad_f=J_g^T*lambda convention。操作=lagrange-boundary；明示資料={\"mode\":\"infeasible_point\",\"model\":{\"constraint_gradient\":[1,3],\"constraint_rhs\":2,\"point\":[3,0]}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(infeasible_point, [1, 3], 2, [3, 0], 3, 否, stationarity_never_repairs_missing_feasibility)","explanation_zh":"Wave6第十三層核對=(infeasible_point, [1, 3], 2, [3, 0], 3, 否, stationarity_never_repairs_missing_feasibility)；依完整明示資料精確重算；regular equality-constrained local extremum只得到必要multiplier condition；stationarity非充分；rank deficient是不適用而非無極值，且本條件不是inequality KKT、global或uniqueness證書。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Lagrange乘子條件"]},"variant":{"family_id":"qf-mod50-lagrange-boundary","seed":202611194078,"parameters":{"mode":"infeasible_point","model":{"constraint_gradient":[1,3],"constraint_rhs":2,"point":[3,0]},"domain":"Exact_equality_constraint_Lagrange_condition","operation":"lagrange-boundary","semantic_template":"lagrange-boundary|{\"mode\":\"infeasible_point\",\"model\":{\"constraint_gradient\":[[1,1],[3,1]],\"constraint_rhs\":[2,1],\"point\":[[3,1],[0,1]]}}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r04-0079","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-multiplier-condition"],"title_zh":"Lagrange乘子條件","stem_zh":"feasibility、constraint-gradient full row rank與grad_f=J_g^T*lambda convention。操作=lagrange-boundary；明示資料={\"mode\":\"kkt_confusion\",\"model\":{\"constraint_kind\":\"inequality\"}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(kkt_confusion, inequality, 否, equality_only_Lagrange_condition_does_not_supply_KKT_sign_or_complementarity_conditions)","explanation_zh":"Wave6第十三層核對=(kkt_confusion, inequality, 否, equality_only_Lagrange_condition_does_not_supply_KKT_sign_or_complementarity_conditions)；依完整明示資料精確重算；regular equality-constrained local extremum只得到必要multiplier condition；stationarity非充分；rank deficient是不適用而非無極值，且本條件不是inequality KKT、global或uniqueness證書。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Lagrange乘子條件"]},"variant":{"family_id":"qf-mod50-lagrange-boundary","seed":202611194079,"parameters":{"mode":"kkt_confusion","model":{"constraint_kind":"inequality"},"domain":"Exact_equality_constraint_Lagrange_condition","operation":"lagrange-boundary","semantic_template":"lagrange-boundary|{\"mode\":\"kkt_confusion\",\"model\":{\"constraint_kind\":\"inequality\"}}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r04-0080","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-multiplier-condition"],"title_zh":"Lagrange乘子條件","stem_zh":"feasibility、constraint-gradient full row rank與grad_f=J_g^T*lambda convention。操作=lagrange-boundary；明示資料={\"mode\":\"rank_deficient\",\"model\":{\"constraint_gradients\":[[1,5],[2,10]]}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(rank_deficient, [[1, 5], [2, 10]], 1, 2, 否, regular_Lagrange_theorem_not_applicable_does_not_mean_no_extremum)","explanation_zh":"Wave6第十三層核對=(rank_deficient, [[1, 5], [2, 10]], 1, 2, 否, regular_Lagrange_theorem_not_applicable_does_not_mean_no_extremum)；依完整明示資料精確重算；regular equality-constrained local extremum只得到必要multiplier condition；stationarity非充分；rank deficient是不適用而非無極值，且本條件不是inequality KKT、global或uniqueness證書。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Lagrange乘子條件"]},"variant":{"family_id":"qf-mod50-lagrange-boundary","seed":202611194080,"parameters":{"mode":"rank_deficient","model":{"constraint_gradients":[[1,5],[2,10]]},"domain":"Exact_equality_constraint_Lagrange_condition","operation":"lagrange-boundary","semantic_template":"lagrange-boundary|{\"mode\":\"rank_deficient\",\"model\":{\"constraint_gradients\":[[[1,1],[5,1]],[[2,1],[10,1]]]}}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r04-0081","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-multiplier-condition"],"title_zh":"Lagrange乘子條件","stem_zh":"feasibility、constraint-gradient full row rank與grad_f=J_g^T*lambda convention。操作=lagrange-boundary；明示資料={\"mode\":\"stationary_not_sufficient\",\"model\":{\"scale\":6}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(stationary_not_sufficient, f(x,y)=-6*x^2, g(x,y)=y=0, [0,0], grad_f=[0,0]=0*grad_g, 是, strict_local_maximum_on_constraint, 否, first_order_multiplier_equation_is_necessary_not_sufficient_for_a_minimum)","explanation_zh":"Wave6第十三層核對=(stationary_not_sufficient, f(x,y)=-6*x^2, g(x,y)=y=0, [0,0], grad_f=[0,0]=0*grad_g, 是, strict_local_maximum_on_constraint, 否, first_order_multiplier_equation_is_necessary_not_sufficient_for_a_minimum)；依完整明示資料精確重算；regular equality-constrained local extremum只得到必要multiplier condition；stationarity非充分；rank deficient是不適用而非無極值，且本條件不是inequality KKT、global或uniqueness證書。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Lagrange乘子條件"]},"variant":{"family_id":"qf-mod50-lagrange-boundary","seed":202611194081,"parameters":{"mode":"stationary_not_sufficient","model":{"scale":6},"domain":"Exact_equality_constraint_Lagrange_condition","operation":"lagrange-boundary","semantic_template":"lagrange-boundary|{\"mode\":\"stationary_not_sufficient\",\"model\":{\"scale\":[6,1]}}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r04-0082","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-multiplier-condition"],"title_zh":"Lagrange乘子條件","stem_zh":"feasibility、constraint-gradient full row rank與grad_f=J_g^T*lambda convention。操作=lagrange-boundary；明示資料={\"mode\":\"infeasible_point\",\"model\":{\"constraint_gradient\":[1,7],\"constraint_rhs\":6,\"point\":[7,0]}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(infeasible_point, [1, 7], 6, [7, 0], 7, 否, stationarity_never_repairs_missing_feasibility)","explanation_zh":"Wave6第十三層核對=(infeasible_point, [1, 7], 6, [7, 0], 7, 否, stationarity_never_repairs_missing_feasibility)；依完整明示資料精確重算；regular equality-constrained local extremum只得到必要multiplier condition；stationarity非充分；rank deficient是不適用而非無極值，且本條件不是inequality KKT、global或uniqueness證書。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Lagrange乘子條件"]},"variant":{"family_id":"qf-mod50-lagrange-boundary","seed":202611194082,"parameters":{"mode":"infeasible_point","model":{"constraint_gradient":[1,7],"constraint_rhs":6,"point":[7,0]},"domain":"Exact_equality_constraint_Lagrange_condition","operation":"lagrange-boundary","semantic_template":"lagrange-boundary|{\"mode\":\"infeasible_point\",\"model\":{\"constraint_gradient\":[[1,1],[7,1]],\"constraint_rhs\":[6,1],\"point\":[[7,1],[0,1]]}}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r04-0083","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-multiplier-condition"],"title_zh":"Lagrange乘子條件","stem_zh":"feasibility、constraint-gradient full row rank與grad_f=J_g^T*lambda convention。操作=lagrange-boundary；明示資料={\"mode\":\"rank_deficient\",\"model\":{\"constraint_gradients\":[[1,8],[2,16]]}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(rank_deficient, [[1, 8], [2, 16]], 1, 2, 否, regular_Lagrange_theorem_not_applicable_does_not_mean_no_extremum)","explanation_zh":"Wave6第十三層核對=(rank_deficient, [[1, 8], [2, 16]], 1, 2, 否, regular_Lagrange_theorem_not_applicable_does_not_mean_no_extremum)；依完整明示資料精確重算；regular equality-constrained local extremum只得到必要multiplier condition；stationarity非充分；rank deficient是不適用而非無極值，且本條件不是inequality KKT、global或uniqueness證書。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Lagrange乘子條件"]},"variant":{"family_id":"qf-mod50-lagrange-boundary","seed":202611194083,"parameters":{"mode":"rank_deficient","model":{"constraint_gradients":[[1,8],[2,16]]},"domain":"Exact_equality_constraint_Lagrange_condition","operation":"lagrange-boundary","semantic_template":"lagrange-boundary|{\"mode\":\"rank_deficient\",\"model\":{\"constraint_gradients\":[[[1,1],[8,1]],[[2,1],[16,1]]]}}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r04-0084","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-multiplier-condition"],"title_zh":"Lagrange乘子條件","stem_zh":"feasibility、constraint-gradient full row rank與grad_f=J_g^T*lambda convention。操作=lagrange-boundary；明示資料={\"mode\":\"rank_deficient\",\"model\":{\"constraint_gradients\":[[1,9],[2,18]]}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(rank_deficient, [[1, 9], [2, 18]], 1, 2, 否, regular_Lagrange_theorem_not_applicable_does_not_mean_no_extremum)","explanation_zh":"Wave6第十三層核對=(rank_deficient, [[1, 9], [2, 18]], 1, 2, 否, regular_Lagrange_theorem_not_applicable_does_not_mean_no_extremum)；依完整明示資料精確重算；regular equality-constrained local extremum只得到必要multiplier condition；stationarity非充分；rank deficient是不適用而非無極值，且本條件不是inequality KKT、global或uniqueness證書。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Lagrange乘子條件"]},"variant":{"family_id":"qf-mod50-lagrange-boundary","seed":202611194084,"parameters":{"mode":"rank_deficient","model":{"constraint_gradients":[[1,9],[2,18]]},"domain":"Exact_equality_constraint_Lagrange_condition","operation":"lagrange-boundary","semantic_template":"lagrange-boundary|{\"mode\":\"rank_deficient\",\"model\":{\"constraint_gradients\":[[[1,1],[9,1]],[[2,1],[18,1]]]}}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r04-0085","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-multiplier-condition"],"title_zh":"Lagrange乘子條件","stem_zh":"feasibility、constraint-gradient full row rank與grad_f=J_g^T*lambda convention。操作=lagrange-boundary；明示資料={\"mode\":\"stationary_not_sufficient\",\"model\":{\"scale\":10}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(stationary_not_sufficient, f(x,y)=-10*x^2, g(x,y)=y=0, [0,0], grad_f=[0,0]=0*grad_g, 是, strict_local_maximum_on_constraint, 否, first_order_multiplier_equation_is_necessary_not_sufficient_for_a_minimum)","explanation_zh":"Wave6第十三層核對=(stationary_not_sufficient, f(x,y)=-10*x^2, g(x,y)=y=0, [0,0], grad_f=[0,0]=0*grad_g, 是, strict_local_maximum_on_constraint, 否, first_order_multiplier_equation_is_necessary_not_sufficient_for_a_minimum)；依完整明示資料精確重算；regular equality-constrained local extremum只得到必要multiplier condition；stationarity非充分；rank deficient是不適用而非無極值，且本條件不是inequality KKT、global或uniqueness證書。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Lagrange乘子條件"]},"variant":{"family_id":"qf-mod50-lagrange-boundary","seed":202611194085,"parameters":{"mode":"stationary_not_sufficient","model":{"scale":10},"domain":"Exact_equality_constraint_Lagrange_condition","operation":"lagrange-boundary","semantic_template":"lagrange-boundary|{\"mode\":\"stationary_not_sufficient\",\"model\":{\"scale\":[10,1]}}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r04-0086","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-multiplier-condition"],"title_zh":"Lagrange乘子條件","stem_zh":"feasibility、constraint-gradient full row rank與grad_f=J_g^T*lambda convention。操作=lagrange-boundary；明示資料={\"mode\":\"infeasible_point\",\"model\":{\"constraint_gradient\":[1,11],\"constraint_rhs\":10,\"point\":[11,0]}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(infeasible_point, [1, 11], 10, [11, 0], 11, 否, stationarity_never_repairs_missing_feasibility)","explanation_zh":"Wave6第十三層核對=(infeasible_point, [1, 11], 10, [11, 0], 11, 否, stationarity_never_repairs_missing_feasibility)；依完整明示資料精確重算；regular equality-constrained local extremum只得到必要multiplier condition；stationarity非充分；rank deficient是不適用而非無極值，且本條件不是inequality KKT、global或uniqueness證書。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Lagrange乘子條件"]},"variant":{"family_id":"qf-mod50-lagrange-boundary","seed":202611194086,"parameters":{"mode":"infeasible_point","model":{"constraint_gradient":[1,11],"constraint_rhs":10,"point":[11,0]},"domain":"Exact_equality_constraint_Lagrange_condition","operation":"lagrange-boundary","semantic_template":"lagrange-boundary|{\"mode\":\"infeasible_point\",\"model\":{\"constraint_gradient\":[[1,1],[11,1]],\"constraint_rhs\":[10,1],\"point\":[[11,1],[0,1]]}}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r04-0087","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-multiplier-condition"],"title_zh":"Lagrange乘子條件","stem_zh":"feasibility、constraint-gradient full row rank與grad_f=J_g^T*lambda convention。操作=lagrange-boundary；明示資料={\"mode\":\"rank_deficient\",\"model\":{\"constraint_gradients\":[[1,12],[2,24]]}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(rank_deficient, [[1, 12], [2, 24]], 1, 2, 否, regular_Lagrange_theorem_not_applicable_does_not_mean_no_extremum)","explanation_zh":"Wave6第十三層核對=(rank_deficient, [[1, 12], [2, 24]], 1, 2, 否, regular_Lagrange_theorem_not_applicable_does_not_mean_no_extremum)；依完整明示資料精確重算；regular equality-constrained local extremum只得到必要multiplier condition；stationarity非充分；rank deficient是不適用而非無極值，且本條件不是inequality KKT、global或uniqueness證書。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Lagrange乘子條件"]},"variant":{"family_id":"qf-mod50-lagrange-boundary","seed":202611194087,"parameters":{"mode":"rank_deficient","model":{"constraint_gradients":[[1,12],[2,24]]},"domain":"Exact_equality_constraint_Lagrange_condition","operation":"lagrange-boundary","semantic_template":"lagrange-boundary|{\"mode\":\"rank_deficient\",\"model\":{\"constraint_gradients\":[[[1,1],[12,1]],[[2,1],[24,1]]]}}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r04-0088","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-multiplier-condition"],"title_zh":"Lagrange乘子條件","stem_zh":"feasibility、constraint-gradient full row rank與grad_f=J_g^T*lambda convention。操作=lagrange-boundary；明示資料={\"mode\":\"rank_deficient\",\"model\":{\"constraint_gradients\":[[1,13],[2,26]]}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(rank_deficient, [[1, 13], [2, 26]], 1, 2, 否, regular_Lagrange_theorem_not_applicable_does_not_mean_no_extremum)","explanation_zh":"Wave6第十三層核對=(rank_deficient, [[1, 13], [2, 26]], 1, 2, 否, regular_Lagrange_theorem_not_applicable_does_not_mean_no_extremum)；依完整明示資料精確重算；regular equality-constrained local extremum只得到必要multiplier condition；stationarity非充分；rank deficient是不適用而非無極值，且本條件不是inequality KKT、global或uniqueness證書。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Lagrange乘子條件"]},"variant":{"family_id":"qf-mod50-lagrange-boundary","seed":202611194088,"parameters":{"mode":"rank_deficient","model":{"constraint_gradients":[[1,13],[2,26]]},"domain":"Exact_equality_constraint_Lagrange_condition","operation":"lagrange-boundary","semantic_template":"lagrange-boundary|{\"mode\":\"rank_deficient\",\"model\":{\"constraint_gradients\":[[[1,1],[13,1]],[[2,1],[26,1]]]}}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r04-0089","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-multiplier-condition"],"title_zh":"Lagrange乘子條件","stem_zh":"feasibility、constraint-gradient full row rank與grad_f=J_g^T*lambda convention。操作=lagrange-boundary；明示資料={\"mode\":\"stationary_not_sufficient\",\"model\":{\"scale\":14}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(stationary_not_sufficient, f(x,y)=-14*x^2, g(x,y)=y=0, [0,0], grad_f=[0,0]=0*grad_g, 是, strict_local_maximum_on_constraint, 否, first_order_multiplier_equation_is_necessary_not_sufficient_for_a_minimum)","explanation_zh":"Wave6第十三層核對=(stationary_not_sufficient, f(x,y)=-14*x^2, g(x,y)=y=0, [0,0], grad_f=[0,0]=0*grad_g, 是, strict_local_maximum_on_constraint, 否, first_order_multiplier_equation_is_necessary_not_sufficient_for_a_minimum)；依完整明示資料精確重算；regular equality-constrained local extremum只得到必要multiplier condition；stationarity非充分；rank deficient是不適用而非無極值，且本條件不是inequality KKT、global或uniqueness證書。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Lagrange乘子條件"]},"variant":{"family_id":"qf-mod50-lagrange-boundary","seed":202611194089,"parameters":{"mode":"stationary_not_sufficient","model":{"scale":14},"domain":"Exact_equality_constraint_Lagrange_condition","operation":"lagrange-boundary","semantic_template":"lagrange-boundary|{\"mode\":\"stationary_not_sufficient\",\"model\":{\"scale\":[14,1]}}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r04-0090","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-multiplier-condition"],"title_zh":"Lagrange乘子條件","stem_zh":"feasibility、constraint-gradient full row rank與grad_f=J_g^T*lambda convention。操作=lagrange-boundary；明示資料={\"mode\":\"infeasible_point\",\"model\":{\"constraint_gradient\":[1,15],\"constraint_rhs\":14,\"point\":[15,0]}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(infeasible_point, [1, 15], 14, [15, 0], 15, 否, stationarity_never_repairs_missing_feasibility)","explanation_zh":"Wave6第十三層核對=(infeasible_point, [1, 15], 14, [15, 0], 15, 否, stationarity_never_repairs_missing_feasibility)；依完整明示資料精確重算；regular equality-constrained local extremum只得到必要multiplier condition；stationarity非充分；rank deficient是不適用而非無極值，且本條件不是inequality KKT、global或uniqueness證書。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Lagrange乘子條件"]},"variant":{"family_id":"qf-mod50-lagrange-boundary","seed":202611194090,"parameters":{"mode":"infeasible_point","model":{"constraint_gradient":[1,15],"constraint_rhs":14,"point":[15,0]},"domain":"Exact_equality_constraint_Lagrange_condition","operation":"lagrange-boundary","semantic_template":"lagrange-boundary|{\"mode\":\"infeasible_point\",\"model\":{\"constraint_gradient\":[[1,1],[15,1]],\"constraint_rhs\":[14,1],\"point\":[[15,1],[0,1]]}}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r04-0091","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-multiplier-condition"],"title_zh":"Lagrange乘子條件","stem_zh":"feasibility、constraint-gradient full row rank與grad_f=J_g^T*lambda convention。操作=lagrange-boundary；明示資料={\"mode\":\"rank_deficient\",\"model\":{\"constraint_gradients\":[[1,16],[2,32]]}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(rank_deficient, [[1, 16], [2, 32]], 1, 2, 否, regular_Lagrange_theorem_not_applicable_does_not_mean_no_extremum)","explanation_zh":"Wave6第十三層核對=(rank_deficient, [[1, 16], [2, 32]], 1, 2, 否, regular_Lagrange_theorem_not_applicable_does_not_mean_no_extremum)；依完整明示資料精確重算；regular equality-constrained local extremum只得到必要multiplier condition；stationarity非充分；rank deficient是不適用而非無極值，且本條件不是inequality KKT、global或uniqueness證書。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Lagrange乘子條件"]},"variant":{"family_id":"qf-mod50-lagrange-boundary","seed":202611194091,"parameters":{"mode":"rank_deficient","model":{"constraint_gradients":[[1,16],[2,32]]},"domain":"Exact_equality_constraint_Lagrange_condition","operation":"lagrange-boundary","semantic_template":"lagrange-boundary|{\"mode\":\"rank_deficient\",\"model\":{\"constraint_gradients\":[[[1,1],[16,1]],[[2,1],[32,1]]]}}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r04-0092","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-multiplier-condition"],"title_zh":"Lagrange乘子條件","stem_zh":"feasibility、constraint-gradient full row rank與grad_f=J_g^T*lambda convention。操作=lagrange-boundary；明示資料={\"mode\":\"rank_deficient\",\"model\":{\"constraint_gradients\":[[1,17],[2,34]]}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(rank_deficient, [[1, 17], [2, 34]], 1, 2, 否, regular_Lagrange_theorem_not_applicable_does_not_mean_no_extremum)","explanation_zh":"Wave6第十三層核對=(rank_deficient, [[1, 17], [2, 34]], 1, 2, 否, regular_Lagrange_theorem_not_applicable_does_not_mean_no_extremum)；依完整明示資料精確重算；regular equality-constrained local extremum只得到必要multiplier condition；stationarity非充分；rank deficient是不適用而非無極值，且本條件不是inequality KKT、global或uniqueness證書。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Lagrange乘子條件"]},"variant":{"family_id":"qf-mod50-lagrange-boundary","seed":202611194092,"parameters":{"mode":"rank_deficient","model":{"constraint_gradients":[[1,17],[2,34]]},"domain":"Exact_equality_constraint_Lagrange_condition","operation":"lagrange-boundary","semantic_template":"lagrange-boundary|{\"mode\":\"rank_deficient\",\"model\":{\"constraint_gradients\":[[[1,1],[17,1]],[[2,1],[34,1]]]}}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r04-0093","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-multiplier-condition"],"title_zh":"Lagrange乘子條件","stem_zh":"feasibility、constraint-gradient full row rank與grad_f=J_g^T*lambda convention。操作=lagrange-boundary；明示資料={\"mode\":\"stationary_not_sufficient\",\"model\":{\"scale\":18}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(stationary_not_sufficient, f(x,y)=-18*x^2, g(x,y)=y=0, [0,0], grad_f=[0,0]=0*grad_g, 是, strict_local_maximum_on_constraint, 否, first_order_multiplier_equation_is_necessary_not_sufficient_for_a_minimum)","explanation_zh":"Wave6第十三層核對=(stationary_not_sufficient, f(x,y)=-18*x^2, g(x,y)=y=0, [0,0], grad_f=[0,0]=0*grad_g, 是, strict_local_maximum_on_constraint, 否, first_order_multiplier_equation_is_necessary_not_sufficient_for_a_minimum)；依完整明示資料精確重算；regular equality-constrained local extremum只得到必要multiplier condition；stationarity非充分；rank deficient是不適用而非無極值，且本條件不是inequality KKT、global或uniqueness證書。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Lagrange乘子條件"]},"variant":{"family_id":"qf-mod50-lagrange-boundary","seed":202611194093,"parameters":{"mode":"stationary_not_sufficient","model":{"scale":18},"domain":"Exact_equality_constraint_Lagrange_condition","operation":"lagrange-boundary","semantic_template":"lagrange-boundary|{\"mode\":\"stationary_not_sufficient\",\"model\":{\"scale\":[18,1]}}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r04-0094","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-multiplier-condition"],"title_zh":"Lagrange乘子條件","stem_zh":"feasibility、constraint-gradient full row rank與grad_f=J_g^T*lambda convention。操作=lagrange-boundary；明示資料={\"mode\":\"infeasible_point\",\"model\":{\"constraint_gradient\":[1,19],\"constraint_rhs\":18,\"point\":[19,0]}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(infeasible_point, [1, 19], 18, [19, 0], 19, 否, stationarity_never_repairs_missing_feasibility)","explanation_zh":"Wave6第十三層核對=(infeasible_point, [1, 19], 18, [19, 0], 19, 否, stationarity_never_repairs_missing_feasibility)；依完整明示資料精確重算；regular equality-constrained local extremum只得到必要multiplier condition；stationarity非充分；rank deficient是不適用而非無極值，且本條件不是inequality KKT、global或uniqueness證書。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Lagrange乘子條件"]},"variant":{"family_id":"qf-mod50-lagrange-boundary","seed":202611194094,"parameters":{"mode":"infeasible_point","model":{"constraint_gradient":[1,19],"constraint_rhs":18,"point":[19,0]},"domain":"Exact_equality_constraint_Lagrange_condition","operation":"lagrange-boundary","semantic_template":"lagrange-boundary|{\"mode\":\"infeasible_point\",\"model\":{\"constraint_gradient\":[[1,1],[19,1]],\"constraint_rhs\":[18,1],\"point\":[[19,1],[0,1]]}}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r04-0095","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-multiplier-condition"],"title_zh":"Lagrange乘子條件","stem_zh":"feasibility、constraint-gradient full row rank與grad_f=J_g^T*lambda convention。操作=lagrange-boundary；明示資料={\"mode\":\"rank_deficient\",\"model\":{\"constraint_gradients\":[[1,20],[2,40]]}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(rank_deficient, [[1, 20], [2, 40]], 1, 2, 否, regular_Lagrange_theorem_not_applicable_does_not_mean_no_extremum)","explanation_zh":"Wave6第十三層核對=(rank_deficient, [[1, 20], [2, 40]], 1, 2, 否, regular_Lagrange_theorem_not_applicable_does_not_mean_no_extremum)；依完整明示資料精確重算；regular equality-constrained local extremum只得到必要multiplier condition；stationarity非充分；rank deficient是不適用而非無極值，且本條件不是inequality KKT、global或uniqueness證書。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Lagrange乘子條件"]},"variant":{"family_id":"qf-mod50-lagrange-boundary","seed":202611194095,"parameters":{"mode":"rank_deficient","model":{"constraint_gradients":[[1,20],[2,40]]},"domain":"Exact_equality_constraint_Lagrange_condition","operation":"lagrange-boundary","semantic_template":"lagrange-boundary|{\"mode\":\"rank_deficient\",\"model\":{\"constraint_gradients\":[[[1,1],[20,1]],[[2,1],[40,1]]]}}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r04-0096","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-multiplier-condition"],"title_zh":"Lagrange乘子條件","stem_zh":"feasibility、constraint-gradient full row rank與grad_f=J_g^T*lambda convention。操作=lagrange-boundary；明示資料={\"mode\":\"rank_deficient\",\"model\":{\"constraint_gradients\":[[1,21],[2,42]]}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(rank_deficient, [[1, 21], [2, 42]], 1, 2, 否, regular_Lagrange_theorem_not_applicable_does_not_mean_no_extremum)","explanation_zh":"Wave6第十三層核對=(rank_deficient, [[1, 21], [2, 42]], 1, 2, 否, regular_Lagrange_theorem_not_applicable_does_not_mean_no_extremum)；依完整明示資料精確重算；regular equality-constrained local extremum只得到必要multiplier condition；stationarity非充分；rank deficient是不適用而非無極值，且本條件不是inequality KKT、global或uniqueness證書。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Lagrange乘子條件"]},"variant":{"family_id":"qf-mod50-lagrange-boundary","seed":202611194096,"parameters":{"mode":"rank_deficient","model":{"constraint_gradients":[[1,21],[2,42]]},"domain":"Exact_equality_constraint_Lagrange_condition","operation":"lagrange-boundary","semantic_template":"lagrange-boundary|{\"mode\":\"rank_deficient\",\"model\":{\"constraint_gradients\":[[[1,1],[21,1]],[[2,1],[42,1]]]}}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r04-0097","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-multiplier-condition"],"title_zh":"Lagrange乘子條件","stem_zh":"feasibility、constraint-gradient full row rank與grad_f=J_g^T*lambda convention。操作=lagrange-boundary；明示資料={\"mode\":\"stationary_not_sufficient\",\"model\":{\"scale\":22}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(stationary_not_sufficient, f(x,y)=-22*x^2, g(x,y)=y=0, [0,0], grad_f=[0,0]=0*grad_g, 是, strict_local_maximum_on_constraint, 否, first_order_multiplier_equation_is_necessary_not_sufficient_for_a_minimum)","explanation_zh":"Wave6第十三層核對=(stationary_not_sufficient, f(x,y)=-22*x^2, g(x,y)=y=0, [0,0], grad_f=[0,0]=0*grad_g, 是, strict_local_maximum_on_constraint, 否, first_order_multiplier_equation_is_necessary_not_sufficient_for_a_minimum)；依完整明示資料精確重算；regular equality-constrained local extremum只得到必要multiplier condition；stationarity非充分；rank deficient是不適用而非無極值，且本條件不是inequality KKT、global或uniqueness證書。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Lagrange乘子條件"]},"variant":{"family_id":"qf-mod50-lagrange-boundary","seed":202611194097,"parameters":{"mode":"stationary_not_sufficient","model":{"scale":22},"domain":"Exact_equality_constraint_Lagrange_condition","operation":"lagrange-boundary","semantic_template":"lagrange-boundary|{\"mode\":\"stationary_not_sufficient\",\"model\":{\"scale\":[22,1]}}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r04-0098","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-multiplier-condition"],"title_zh":"Lagrange乘子條件","stem_zh":"feasibility、constraint-gradient full row rank與grad_f=J_g^T*lambda convention。操作=lagrange-boundary；明示資料={\"mode\":\"infeasible_point\",\"model\":{\"constraint_gradient\":[1,23],\"constraint_rhs\":22,\"point\":[23,0]}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(infeasible_point, [1, 23], 22, [23, 0], 23, 否, stationarity_never_repairs_missing_feasibility)","explanation_zh":"Wave6第十三層核對=(infeasible_point, [1, 23], 22, [23, 0], 23, 否, stationarity_never_repairs_missing_feasibility)；依完整明示資料精確重算；regular equality-constrained local extremum只得到必要multiplier condition；stationarity非充分；rank deficient是不適用而非無極值，且本條件不是inequality KKT、global或uniqueness證書。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Lagrange乘子條件"]},"variant":{"family_id":"qf-mod50-lagrange-boundary","seed":202611194098,"parameters":{"mode":"infeasible_point","model":{"constraint_gradient":[1,23],"constraint_rhs":22,"point":[23,0]},"domain":"Exact_equality_constraint_Lagrange_condition","operation":"lagrange-boundary","semantic_template":"lagrange-boundary|{\"mode\":\"infeasible_point\",\"model\":{\"constraint_gradient\":[[1,1],[23,1]],\"constraint_rhs\":[22,1],\"point\":[[23,1],[0,1]]}}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r04-0099","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-multiplier-condition"],"title_zh":"Lagrange乘子條件","stem_zh":"feasibility、constraint-gradient full row rank與grad_f=J_g^T*lambda convention。操作=lagrange-boundary；明示資料={\"mode\":\"rank_deficient\",\"model\":{\"constraint_gradients\":[[1,24],[2,48]]}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(rank_deficient, [[1, 24], [2, 48]], 1, 2, 否, regular_Lagrange_theorem_not_applicable_does_not_mean_no_extremum)","explanation_zh":"Wave6第十三層核對=(rank_deficient, [[1, 24], [2, 48]], 1, 2, 否, regular_Lagrange_theorem_not_applicable_does_not_mean_no_extremum)；依完整明示資料精確重算；regular equality-constrained local extremum只得到必要multiplier condition；stationarity非充分；rank deficient是不適用而非無極值，且本條件不是inequality KKT、global或uniqueness證書。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Lagrange乘子條件"]},"variant":{"family_id":"qf-mod50-lagrange-boundary","seed":202611194099,"parameters":{"mode":"rank_deficient","model":{"constraint_gradients":[[1,24],[2,48]]},"domain":"Exact_equality_constraint_Lagrange_condition","operation":"lagrange-boundary","semantic_template":"lagrange-boundary|{\"mode\":\"rank_deficient\",\"model\":{\"constraint_gradients\":[[[1,1],[24,1]],[[2,1],[48,1]]]}}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-pulse-20261119-r04-0100","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","language":"zh-Hant","source_mko_ids":["mko-lagrange-multiplier-condition"],"title_zh":"Lagrange乘子條件","stem_zh":"feasibility、constraint-gradient full row rank與grad_f=J_g^T*lambda convention。操作=lagrange-boundary；明示資料={\"mode\":\"rank_deficient\",\"model\":{\"constraint_gradients\":[[1,25],[2,50]]}}；請依 exact_ordered_tuple_declared_by_interface 回答。","answer":{"kind":"expression","value":"(rank_deficient, [[1, 25], [2, 50]], 1, 2, 否, regular_Lagrange_theorem_not_applicable_does_not_mean_no_extremum)","explanation_zh":"Wave6第十三層核對=(rank_deficient, [[1, 25], [2, 50]], 1, 2, 否, regular_Lagrange_theorem_not_applicable_does_not_mean_no_extremum)；依完整明示資料精確重算；regular equality-constrained local extremum只得到必要multiplier condition；stationarity非充分；rank deficient是不適用而非無極值，且本條件不是inequality KKT、global或uniqueness證書。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Lagrange乘子條件"]},"variant":{"family_id":"qf-mod50-lagrange-boundary","seed":202611194100,"parameters":{"mode":"rank_deficient","model":{"constraint_gradients":[[1,25],[2,50]]},"domain":"Exact_equality_constraint_Lagrange_condition","operation":"lagrange-boundary","semantic_template":"lagrange-boundary|{\"mode\":\"rank_deficient\",\"model\":{\"constraint_gradients\":[[[1,1],[25,1]],[[2,1],[50,1]]]}}"}},"verification":{"status":"candidate","checks":["producer_contract_recomputation","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"candidate 僅對明示有限或符號模型重算；不構成一般定理、人工審定、Lean 或 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"GPT-5.6 Terra","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}