{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r04-linear-program-duality-0001","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-program-duality"],"title_zh":"線性規劃對偶","stem_zh":"固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-construction-qualification；資料={\"A\":[[1,0],[0,1]],\"b\":[2,3],\"c\":[1,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 2, [[1, 0], [0, 1]], [2, 3], [1, 1], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, 是, dimensions_signs_and_transpose_are_fixed_by_one_convention)","explanation_zh":"Wave6第九層核對=(2, 2, [[1, 0], [0, 1]], [2, 3], [1, 1], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, 是, dimensions_signs_and_transpose_are_fixed_by_one_convention)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性規劃對偶"]},"variant":{"family_id":"qf-mod46-lpdual-construction-qualification","seed":202611154001,"parameters":{"A":[[1,0],[0,1]],"b":[2,3],"c":[1,1],"domain":"Exact_primal_dual_linear_program","operation":"lpdual-construction-qualification","semantic_template":"qf-mod46-lpdual-construction-qualification/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[2,1],[3,1]],\"c\":[[1,1],[1,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r04-linear-program-duality-0002","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-program-duality"],"title_zh":"線性規劃對偶","stem_zh":"固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-construction-qualification；資料={\"A\":[[1,0],[0,1]],\"b\":[3,4],\"c\":[2,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 2, [[1, 0], [0, 1]], [3, 4], [2, 2], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, 是, dimensions_signs_and_transpose_are_fixed_by_one_convention)","explanation_zh":"Wave6第九層核對=(2, 2, [[1, 0], [0, 1]], [3, 4], [2, 2], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, 是, dimensions_signs_and_transpose_are_fixed_by_one_convention)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性規劃對偶"]},"variant":{"family_id":"qf-mod46-lpdual-construction-qualification","seed":202611154002,"parameters":{"A":[[1,0],[0,1]],"b":[3,4],"c":[2,2],"domain":"Exact_primal_dual_linear_program","operation":"lpdual-construction-qualification","semantic_template":"qf-mod46-lpdual-construction-qualification/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[3,1],[4,1]],\"c\":[[2,1],[2,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r04-linear-program-duality-0003","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-program-duality"],"title_zh":"線性規劃對偶","stem_zh":"固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-construction-qualification；資料={\"A\":[[1,0],[0,1]],\"b\":[4,5],\"c\":[3,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 2, [[1, 0], [0, 1]], [4, 5], [3, 3], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, 是, dimensions_signs_and_transpose_are_fixed_by_one_convention)","explanation_zh":"Wave6第九層核對=(2, 2, [[1, 0], [0, 1]], [4, 5], [3, 3], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, 是, dimensions_signs_and_transpose_are_fixed_by_one_convention)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性規劃對偶"]},"variant":{"family_id":"qf-mod46-lpdual-construction-qualification","seed":202611154003,"parameters":{"A":[[1,0],[0,1]],"b":[4,5],"c":[3,3],"domain":"Exact_primal_dual_linear_program","operation":"lpdual-construction-qualification","semantic_template":"qf-mod46-lpdual-construction-qualification/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[4,1],[5,1]],\"c\":[[3,1],[3,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r04-linear-program-duality-0004","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-program-duality"],"title_zh":"線性規劃對偶","stem_zh":"固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-construction-qualification；資料={\"A\":[[1,0],[0,1]],\"b\":[5,6],\"c\":[4,4]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 2, [[1, 0], [0, 1]], [5, 6], [4, 4], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, 是, dimensions_signs_and_transpose_are_fixed_by_one_convention)","explanation_zh":"Wave6第九層核對=(2, 2, [[1, 0], [0, 1]], [5, 6], [4, 4], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, 是, dimensions_signs_and_transpose_are_fixed_by_one_convention)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性規劃對偶"]},"variant":{"family_id":"qf-mod46-lpdual-construction-qualification","seed":202611154004,"parameters":{"A":[[1,0],[0,1]],"b":[5,6],"c":[4,4],"domain":"Exact_primal_dual_linear_program","operation":"lpdual-construction-qualification","semantic_template":"qf-mod46-lpdual-construction-qualification/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[5,1],[6,1]],\"c\":[[4,1],[4,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r04-linear-program-duality-0005","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-program-duality"],"title_zh":"線性規劃對偶","stem_zh":"固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-construction-qualification；資料={\"A\":[[1,0],[0,1]],\"b\":[6,7],\"c\":[1,5]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 2, [[1, 0], [0, 1]], [6, 7], [1, 5], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, 是, dimensions_signs_and_transpose_are_fixed_by_one_convention)","explanation_zh":"Wave6第九層核對=(2, 2, [[1, 0], [0, 1]], [6, 7], [1, 5], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, 是, dimensions_signs_and_transpose_are_fixed_by_one_convention)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性規劃對偶"]},"variant":{"family_id":"qf-mod46-lpdual-construction-qualification","seed":202611154005,"parameters":{"A":[[1,0],[0,1]],"b":[6,7],"c":[1,5],"domain":"Exact_primal_dual_linear_program","operation":"lpdual-construction-qualification","semantic_template":"qf-mod46-lpdual-construction-qualification/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[6,1],[7,1]],\"c\":[[1,1],[5,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r04-linear-program-duality-0006","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-program-duality"],"title_zh":"線性規劃對偶","stem_zh":"固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-construction-qualification；資料={\"A\":[[1,0],[0,1]],\"b\":[7,8],\"c\":[2,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 2, [[1, 0], [0, 1]], [7, 8], [2, 1], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, 是, dimensions_signs_and_transpose_are_fixed_by_one_convention)","explanation_zh":"Wave6第九層核對=(2, 2, [[1, 0], [0, 1]], [7, 8], [2, 1], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, 是, dimensions_signs_and_transpose_are_fixed_by_one_convention)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性規劃對偶"]},"variant":{"family_id":"qf-mod46-lpdual-construction-qualification","seed":202611154006,"parameters":{"A":[[1,0],[0,1]],"b":[7,8],"c":[2,1],"domain":"Exact_primal_dual_linear_program","operation":"lpdual-construction-qualification","semantic_template":"qf-mod46-lpdual-construction-qualification/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[7,1],[8,1]],\"c\":[[2,1],[1,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r04-linear-program-duality-0007","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-program-duality"],"title_zh":"線性規劃對偶","stem_zh":"固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-construction-qualification；資料={\"A\":[[1,0],[0,1]],\"b\":[8,9],\"c\":[3,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 2, [[1, 0], [0, 1]], [8, 9], [3, 2], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, 是, dimensions_signs_and_transpose_are_fixed_by_one_convention)","explanation_zh":"Wave6第九層核對=(2, 2, [[1, 0], [0, 1]], [8, 9], [3, 2], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, 是, dimensions_signs_and_transpose_are_fixed_by_one_convention)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性規劃對偶"]},"variant":{"family_id":"qf-mod46-lpdual-construction-qualification","seed":202611154007,"parameters":{"A":[[1,0],[0,1]],"b":[8,9],"c":[3,2],"domain":"Exact_primal_dual_linear_program","operation":"lpdual-construction-qualification","semantic_template":"qf-mod46-lpdual-construction-qualification/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[8,1],[9,1]],\"c\":[[3,1],[2,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r04-linear-program-duality-0008","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-program-duality"],"title_zh":"線性規劃對偶","stem_zh":"固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-construction-qualification；資料={\"A\":[[1,0],[0,1]],\"b\":[9,10],\"c\":[4,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 2, [[1, 0], [0, 1]], [9, 10], [4, 3], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, 是, dimensions_signs_and_transpose_are_fixed_by_one_convention)","explanation_zh":"Wave6第九層核對=(2, 2, [[1, 0], [0, 1]], [9, 10], [4, 3], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, 是, dimensions_signs_and_transpose_are_fixed_by_one_convention)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性規劃對偶"]},"variant":{"family_id":"qf-mod46-lpdual-construction-qualification","seed":202611154008,"parameters":{"A":[[1,0],[0,1]],"b":[9,10],"c":[4,3],"domain":"Exact_primal_dual_linear_program","operation":"lpdual-construction-qualification","semantic_template":"qf-mod46-lpdual-construction-qualification/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[9,1],[10,1]],\"c\":[[4,1],[3,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r04-linear-program-duality-0009","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-program-duality"],"title_zh":"線性規劃對偶","stem_zh":"固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-construction-qualification；資料={\"A\":[[1,0],[0,1]],\"b\":[10,11],\"c\":[1,4]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 2, [[1, 0], [0, 1]], [10, 11], [1, 4], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, 是, dimensions_signs_and_transpose_are_fixed_by_one_convention)","explanation_zh":"Wave6第九層核對=(2, 2, [[1, 0], [0, 1]], [10, 11], [1, 4], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, 是, dimensions_signs_and_transpose_are_fixed_by_one_convention)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性規劃對偶"]},"variant":{"family_id":"qf-mod46-lpdual-construction-qualification","seed":202611154009,"parameters":{"A":[[1,0],[0,1]],"b":[10,11],"c":[1,4],"domain":"Exact_primal_dual_linear_program","operation":"lpdual-construction-qualification","semantic_template":"qf-mod46-lpdual-construction-qualification/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[10,1],[11,1]],\"c\":[[1,1],[4,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r04-linear-program-duality-0010","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-program-duality"],"title_zh":"線性規劃對偶","stem_zh":"固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-construction-qualification；資料={\"A\":[[1,0],[0,1]],\"b\":[11,12],\"c\":[2,5]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 2, [[1, 0], [0, 1]], [11, 12], [2, 5], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, 是, dimensions_signs_and_transpose_are_fixed_by_one_convention)","explanation_zh":"Wave6第九層核對=(2, 2, [[1, 0], [0, 1]], [11, 12], [2, 5], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, 是, dimensions_signs_and_transpose_are_fixed_by_one_convention)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性規劃對偶"]},"variant":{"family_id":"qf-mod46-lpdual-construction-qualification","seed":202611154010,"parameters":{"A":[[1,0],[0,1]],"b":[11,12],"c":[2,5],"domain":"Exact_primal_dual_linear_program","operation":"lpdual-construction-qualification","semantic_template":"qf-mod46-lpdual-construction-qualification/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[11,1],[12,1]],\"c\":[[2,1],[5,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r04-linear-program-duality-0011","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-program-duality"],"title_zh":"線性規劃對偶","stem_zh":"固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-construction-qualification；資料={\"A\":[[1,0],[0,1]],\"b\":[12,13],\"c\":[3,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 2, [[1, 0], [0, 1]], [12, 13], [3, 1], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, 是, dimensions_signs_and_transpose_are_fixed_by_one_convention)","explanation_zh":"Wave6第九層核對=(2, 2, [[1, 0], [0, 1]], [12, 13], [3, 1], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, 是, dimensions_signs_and_transpose_are_fixed_by_one_convention)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性規劃對偶"]},"variant":{"family_id":"qf-mod46-lpdual-construction-qualification","seed":202611154011,"parameters":{"A":[[1,0],[0,1]],"b":[12,13],"c":[3,1],"domain":"Exact_primal_dual_linear_program","operation":"lpdual-construction-qualification","semantic_template":"qf-mod46-lpdual-construction-qualification/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[12,1],[13,1]],\"c\":[[3,1],[1,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r04-linear-program-duality-0012","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-program-duality"],"title_zh":"線性規劃對偶","stem_zh":"固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-construction-qualification；資料={\"A\":[[1,0],[0,1]],\"b\":[13,14],\"c\":[4,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 2, [[1, 0], [0, 1]], [13, 14], [4, 2], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, 是, dimensions_signs_and_transpose_are_fixed_by_one_convention)","explanation_zh":"Wave6第九層核對=(2, 2, [[1, 0], [0, 1]], [13, 14], [4, 2], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, 是, dimensions_signs_and_transpose_are_fixed_by_one_convention)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性規劃對偶"]},"variant":{"family_id":"qf-mod46-lpdual-construction-qualification","seed":202611154012,"parameters":{"A":[[1,0],[0,1]],"b":[13,14],"c":[4,2],"domain":"Exact_primal_dual_linear_program","operation":"lpdual-construction-qualification","semantic_template":"qf-mod46-lpdual-construction-qualification/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[13,1],[14,1]],\"c\":[[4,1],[2,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r04-linear-program-duality-0013","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-program-duality"],"title_zh":"線性規劃對偶","stem_zh":"固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-construction-qualification；資料={\"A\":[[1,0],[0,1]],\"b\":[14,15],\"c\":[1,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 2, [[1, 0], [0, 1]], [14, 15], [1, 3], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, 是, dimensions_signs_and_transpose_are_fixed_by_one_convention)","explanation_zh":"Wave6第九層核對=(2, 2, [[1, 0], [0, 1]], [14, 15], [1, 3], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, 是, dimensions_signs_and_transpose_are_fixed_by_one_convention)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性規劃對偶"]},"variant":{"family_id":"qf-mod46-lpdual-construction-qualification","seed":202611154013,"parameters":{"A":[[1,0],[0,1]],"b":[14,15],"c":[1,3],"domain":"Exact_primal_dual_linear_program","operation":"lpdual-construction-qualification","semantic_template":"qf-mod46-lpdual-construction-qualification/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[14,1],[15,1]],\"c\":[[1,1],[3,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r04-linear-program-duality-0014","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-program-duality"],"title_zh":"線性規劃對偶","stem_zh":"固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-construction-qualification；資料={\"A\":[[1,0],[0,1]],\"b\":[15,16],\"c\":[2,4]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 2, [[1, 0], [0, 1]], [15, 16], [2, 4], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, 是, dimensions_signs_and_transpose_are_fixed_by_one_convention)","explanation_zh":"Wave6第九層核對=(2, 2, [[1, 0], [0, 1]], [15, 16], [2, 4], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, 是, dimensions_signs_and_transpose_are_fixed_by_one_convention)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性規劃對偶"]},"variant":{"family_id":"qf-mod46-lpdual-construction-qualification","seed":202611154014,"parameters":{"A":[[1,0],[0,1]],"b":[15,16],"c":[2,4],"domain":"Exact_primal_dual_linear_program","operation":"lpdual-construction-qualification","semantic_template":"qf-mod46-lpdual-construction-qualification/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[15,1],[16,1]],\"c\":[[2,1],[4,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r04-linear-program-duality-0015","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-program-duality"],"title_zh":"線性規劃對偶","stem_zh":"固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-construction-qualification；資料={\"A\":[[1,0],[0,1]],\"b\":[16,17],\"c\":[3,5]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 2, [[1, 0], [0, 1]], [16, 17], [3, 5], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, 是, dimensions_signs_and_transpose_are_fixed_by_one_convention)","explanation_zh":"Wave6第九層核對=(2, 2, [[1, 0], [0, 1]], [16, 17], [3, 5], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, 是, dimensions_signs_and_transpose_are_fixed_by_one_convention)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性規劃對偶"]},"variant":{"family_id":"qf-mod46-lpdual-construction-qualification","seed":202611154015,"parameters":{"A":[[1,0],[0,1]],"b":[16,17],"c":[3,5],"domain":"Exact_primal_dual_linear_program","operation":"lpdual-construction-qualification","semantic_template":"qf-mod46-lpdual-construction-qualification/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[16,1],[17,1]],\"c\":[[3,1],[5,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r04-linear-program-duality-0016","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-program-duality"],"title_zh":"線性規劃對偶","stem_zh":"固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-construction-qualification；資料={\"A\":[[1,0],[0,1]],\"b\":[17,18],\"c\":[4,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 2, [[1, 0], [0, 1]], [17, 18], [4, 1], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, 是, dimensions_signs_and_transpose_are_fixed_by_one_convention)","explanation_zh":"Wave6第九層核對=(2, 2, [[1, 0], [0, 1]], [17, 18], [4, 1], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, 是, dimensions_signs_and_transpose_are_fixed_by_one_convention)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性規劃對偶"]},"variant":{"family_id":"qf-mod46-lpdual-construction-qualification","seed":202611154016,"parameters":{"A":[[1,0],[0,1]],"b":[17,18],"c":[4,1],"domain":"Exact_primal_dual_linear_program","operation":"lpdual-construction-qualification","semantic_template":"qf-mod46-lpdual-construction-qualification/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[17,1],[18,1]],\"c\":[[4,1],[1,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r04-linear-program-duality-0017","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-program-duality"],"title_zh":"線性規劃對偶","stem_zh":"固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-construction-qualification；資料={\"A\":[[1,0],[0,1]],\"b\":[18,19],\"c\":[1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 2, [[1, 0], [0, 1]], [18, 19], [1, 2], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, 是, dimensions_signs_and_transpose_are_fixed_by_one_convention)","explanation_zh":"Wave6第九層核對=(2, 2, [[1, 0], [0, 1]], [18, 19], [1, 2], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, 是, dimensions_signs_and_transpose_are_fixed_by_one_convention)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性規劃對偶"]},"variant":{"family_id":"qf-mod46-lpdual-construction-qualification","seed":202611154017,"parameters":{"A":[[1,0],[0,1]],"b":[18,19],"c":[1,2],"domain":"Exact_primal_dual_linear_program","operation":"lpdual-construction-qualification","semantic_template":"qf-mod46-lpdual-construction-qualification/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[18,1],[19,1]],\"c\":[[1,1],[2,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r04-linear-program-duality-0018","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-program-duality"],"title_zh":"線性規劃對偶","stem_zh":"固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-construction-qualification；資料={\"A\":[[1,0],[0,1]],\"b\":[19,20],\"c\":[2,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 2, [[1, 0], [0, 1]], [19, 20], [2, 3], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, 是, dimensions_signs_and_transpose_are_fixed_by_one_convention)","explanation_zh":"Wave6第九層核對=(2, 2, [[1, 0], [0, 1]], [19, 20], [2, 3], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, 是, dimensions_signs_and_transpose_are_fixed_by_one_convention)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性規劃對偶"]},"variant":{"family_id":"qf-mod46-lpdual-construction-qualification","seed":202611154018,"parameters":{"A":[[1,0],[0,1]],"b":[19,20],"c":[2,3],"domain":"Exact_primal_dual_linear_program","operation":"lpdual-construction-qualification","semantic_template":"qf-mod46-lpdual-construction-qualification/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[19,1],[20,1]],\"c\":[[2,1],[3,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r04-linear-program-duality-0019","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-program-duality"],"title_zh":"線性規劃對偶","stem_zh":"固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-construction-qualification；資料={\"A\":[[1,0],[0,1]],\"b\":[20,21],\"c\":[3,4]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 2, [[1, 0], [0, 1]], [20, 21], [3, 4], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, 是, dimensions_signs_and_transpose_are_fixed_by_one_convention)","explanation_zh":"Wave6第九層核對=(2, 2, [[1, 0], [0, 1]], [20, 21], [3, 4], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, 是, dimensions_signs_and_transpose_are_fixed_by_one_convention)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性規劃對偶"]},"variant":{"family_id":"qf-mod46-lpdual-construction-qualification","seed":202611154019,"parameters":{"A":[[1,0],[0,1]],"b":[20,21],"c":[3,4],"domain":"Exact_primal_dual_linear_program","operation":"lpdual-construction-qualification","semantic_template":"qf-mod46-lpdual-construction-qualification/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[20,1],[21,1]],\"c\":[[3,1],[4,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r04-linear-program-duality-0020","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-program-duality"],"title_zh":"線性規劃對偶","stem_zh":"固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-construction-qualification；資料={\"A\":[[1,0],[0,1]],\"b\":[21,22],\"c\":[4,5]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 2, [[1, 0], [0, 1]], [21, 22], [4, 5], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, 是, dimensions_signs_and_transpose_are_fixed_by_one_convention)","explanation_zh":"Wave6第九層核對=(2, 2, [[1, 0], [0, 1]], [21, 22], [4, 5], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, 是, dimensions_signs_and_transpose_are_fixed_by_one_convention)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性規劃對偶"]},"variant":{"family_id":"qf-mod46-lpdual-construction-qualification","seed":202611154020,"parameters":{"A":[[1,0],[0,1]],"b":[21,22],"c":[4,5],"domain":"Exact_primal_dual_linear_program","operation":"lpdual-construction-qualification","semantic_template":"qf-mod46-lpdual-construction-qualification/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[21,1],[22,1]],\"c\":[[4,1],[5,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r04-linear-program-duality-0021","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-program-duality"],"title_zh":"線性規劃對偶","stem_zh":"固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-construction-qualification；資料={\"A\":[[1,0],[0,1]],\"b\":[22,23],\"c\":[1,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 2, [[1, 0], [0, 1]], [22, 23], [1, 1], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, 是, dimensions_signs_and_transpose_are_fixed_by_one_convention)","explanation_zh":"Wave6第九層核對=(2, 2, [[1, 0], [0, 1]], [22, 23], [1, 1], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, 是, dimensions_signs_and_transpose_are_fixed_by_one_convention)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性規劃對偶"]},"variant":{"family_id":"qf-mod46-lpdual-construction-qualification","seed":202611154021,"parameters":{"A":[[1,0],[0,1]],"b":[22,23],"c":[1,1],"domain":"Exact_primal_dual_linear_program","operation":"lpdual-construction-qualification","semantic_template":"qf-mod46-lpdual-construction-qualification/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[22,1],[23,1]],\"c\":[[1,1],[1,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r04-linear-program-duality-0022","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-program-duality"],"title_zh":"線性規劃對偶","stem_zh":"固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-construction-qualification；資料={\"A\":[[1,0],[0,1]],\"b\":[23,24],\"c\":[2,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 2, [[1, 0], [0, 1]], [23, 24], [2, 2], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, 是, dimensions_signs_and_transpose_are_fixed_by_one_convention)","explanation_zh":"Wave6第九層核對=(2, 2, [[1, 0], [0, 1]], [23, 24], [2, 2], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, 是, dimensions_signs_and_transpose_are_fixed_by_one_convention)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性規劃對偶"]},"variant":{"family_id":"qf-mod46-lpdual-construction-qualification","seed":202611154022,"parameters":{"A":[[1,0],[0,1]],"b":[23,24],"c":[2,2],"domain":"Exact_primal_dual_linear_program","operation":"lpdual-construction-qualification","semantic_template":"qf-mod46-lpdual-construction-qualification/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[23,1],[24,1]],\"c\":[[2,1],[2,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r04-linear-program-duality-0023","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-program-duality"],"title_zh":"線性規劃對偶","stem_zh":"固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-construction-qualification；資料={\"A\":[[1,0],[0,1]],\"b\":[24,25],\"c\":[3,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 2, [[1, 0], [0, 1]], [24, 25], [3, 3], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, 是, dimensions_signs_and_transpose_are_fixed_by_one_convention)","explanation_zh":"Wave6第九層核對=(2, 2, [[1, 0], [0, 1]], [24, 25], [3, 3], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, 是, dimensions_signs_and_transpose_are_fixed_by_one_convention)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性規劃對偶"]},"variant":{"family_id":"qf-mod46-lpdual-construction-qualification","seed":202611154023,"parameters":{"A":[[1,0],[0,1]],"b":[24,25],"c":[3,3],"domain":"Exact_primal_dual_linear_program","operation":"lpdual-construction-qualification","semantic_template":"qf-mod46-lpdual-construction-qualification/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[24,1],[25,1]],\"c\":[[3,1],[3,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r04-linear-program-duality-0024","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-program-duality"],"title_zh":"線性規劃對偶","stem_zh":"固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-construction-qualification；資料={\"A\":[[1,0],[0,1]],\"b\":[25,26],\"c\":[4,4]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 2, [[1, 0], [0, 1]], [25, 26], [4, 4], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, 是, dimensions_signs_and_transpose_are_fixed_by_one_convention)","explanation_zh":"Wave6第九層核對=(2, 2, [[1, 0], [0, 1]], [25, 26], [4, 4], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, 是, dimensions_signs_and_transpose_are_fixed_by_one_convention)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性規劃對偶"]},"variant":{"family_id":"qf-mod46-lpdual-construction-qualification","seed":202611154024,"parameters":{"A":[[1,0],[0,1]],"b":[25,26],"c":[4,4],"domain":"Exact_primal_dual_linear_program","operation":"lpdual-construction-qualification","semantic_template":"qf-mod46-lpdual-construction-qualification/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[25,1],[26,1]],\"c\":[[4,1],[4,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r04-linear-program-duality-0025","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-program-duality"],"title_zh":"線性規劃對偶","stem_zh":"固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-construction-qualification；資料={\"A\":[[1,0],[0,1]],\"b\":[26,27],\"c\":[1,5]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 2, [[1, 0], [0, 1]], [26, 27], [1, 5], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, 是, dimensions_signs_and_transpose_are_fixed_by_one_convention)","explanation_zh":"Wave6第九層核對=(2, 2, [[1, 0], [0, 1]], [26, 27], [1, 5], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, 是, dimensions_signs_and_transpose_are_fixed_by_one_convention)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性規劃對偶"]},"variant":{"family_id":"qf-mod46-lpdual-construction-qualification","seed":202611154025,"parameters":{"A":[[1,0],[0,1]],"b":[26,27],"c":[1,5],"domain":"Exact_primal_dual_linear_program","operation":"lpdual-construction-qualification","semantic_template":"qf-mod46-lpdual-construction-qualification/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[26,1],[27,1]],\"c\":[[1,1],[5,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r04-linear-program-duality-0026","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-program-duality"],"title_zh":"線性規劃對偶","stem_zh":"固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-weak-duality-chain；資料={\"A\":[[1,0],[0,1]],\"b\":[2,3],\"c\":[1,1],\"x\":[[2,2],[3,2]],\"y\":[2,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([1, 3/2], [2, 2], [1, 3/2], [2, 2], 是, 是, 5/2, 5, 10, 是, 是, 是, cTx_le_yTAx_le_bTy_for_every_feasible_pair)","explanation_zh":"Wave6第九層核對=([1, 3/2], [2, 2], [1, 3/2], [2, 2], 是, 是, 5/2, 5, 10, 是, 是, 是, cTx_le_yTAx_le_bTy_for_every_feasible_pair)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性規劃對偶"]},"variant":{"family_id":"qf-mod46-lpdual-weak-duality-chain","seed":202611154026,"parameters":{"A":[[1,0],[0,1]],"b":[2,3],"c":[1,1],"x":[[2,2],[3,2]],"y":[2,2],"domain":"Exact_primal_dual_linear_program","operation":"lpdual-weak-duality-chain","semantic_template":"qf-mod46-lpdual-weak-duality-chain/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[2,1],[3,1]],\"c\":[[1,1],[1,1]],\"x\":[[1,1],[3,2]],\"y\":[[2,1],[2,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r04-linear-program-duality-0027","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-program-duality"],"title_zh":"線性規劃對偶","stem_zh":"固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-weak-duality-chain；資料={\"A\":[[1,0],[0,1]],\"b\":[3,4],\"c\":[2,2],\"x\":[[3,2],[4,2]],\"y\":[3,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3/2, 2], [3, 3], [3/2, 2], [3, 3], 是, 是, 7, 21/2, 21, 是, 是, 是, cTx_le_yTAx_le_bTy_for_every_feasible_pair)","explanation_zh":"Wave6第九層核對=([3/2, 2], [3, 3], [3/2, 2], [3, 3], 是, 是, 7, 21/2, 21, 是, 是, 是, cTx_le_yTAx_le_bTy_for_every_feasible_pair)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性規劃對偶"]},"variant":{"family_id":"qf-mod46-lpdual-weak-duality-chain","seed":202611154027,"parameters":{"A":[[1,0],[0,1]],"b":[3,4],"c":[2,2],"x":[[3,2],[4,2]],"y":[3,3],"domain":"Exact_primal_dual_linear_program","operation":"lpdual-weak-duality-chain","semantic_template":"qf-mod46-lpdual-weak-duality-chain/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[3,1],[4,1]],\"c\":[[2,1],[2,1]],\"x\":[[3,2],[2,1]],\"y\":[[3,1],[3,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r04-linear-program-duality-0028","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-program-duality"],"title_zh":"線性規劃對偶","stem_zh":"固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-weak-duality-chain；資料={\"A\":[[1,0],[0,1]],\"b\":[4,5],\"c\":[3,3],\"x\":[[4,2],[5,2]],\"y\":[4,4]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([2, 5/2], [4, 4], [2, 5/2], [4, 4], 是, 是, 27/2, 18, 36, 是, 是, 是, cTx_le_yTAx_le_bTy_for_every_feasible_pair)","explanation_zh":"Wave6第九層核對=([2, 5/2], [4, 4], [2, 5/2], [4, 4], 是, 是, 27/2, 18, 36, 是, 是, 是, cTx_le_yTAx_le_bTy_for_every_feasible_pair)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性規劃對偶"]},"variant":{"family_id":"qf-mod46-lpdual-weak-duality-chain","seed":202611154028,"parameters":{"A":[[1,0],[0,1]],"b":[4,5],"c":[3,3],"x":[[4,2],[5,2]],"y":[4,4],"domain":"Exact_primal_dual_linear_program","operation":"lpdual-weak-duality-chain","semantic_template":"qf-mod46-lpdual-weak-duality-chain/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[4,1],[5,1]],\"c\":[[3,1],[3,1]],\"x\":[[2,1],[5,2]],\"y\":[[4,1],[4,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r04-linear-program-duality-0029","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-program-duality"],"title_zh":"線性規劃對偶","stem_zh":"固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-weak-duality-chain；資料={\"A\":[[1,0],[0,1]],\"b\":[5,6],\"c\":[4,4],\"x\":[[5,2],[6,2]],\"y\":[5,5]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([5/2, 3], [5, 5], [5/2, 3], [5, 5], 是, 是, 22, 55/2, 55, 是, 是, 是, cTx_le_yTAx_le_bTy_for_every_feasible_pair)","explanation_zh":"Wave6第九層核對=([5/2, 3], [5, 5], [5/2, 3], [5, 5], 是, 是, 22, 55/2, 55, 是, 是, 是, cTx_le_yTAx_le_bTy_for_every_feasible_pair)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性規劃對偶"]},"variant":{"family_id":"qf-mod46-lpdual-weak-duality-chain","seed":202611154029,"parameters":{"A":[[1,0],[0,1]],"b":[5,6],"c":[4,4],"x":[[5,2],[6,2]],"y":[5,5],"domain":"Exact_primal_dual_linear_program","operation":"lpdual-weak-duality-chain","semantic_template":"qf-mod46-lpdual-weak-duality-chain/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[5,1],[6,1]],\"c\":[[4,1],[4,1]],\"x\":[[5,2],[3,1]],\"y\":[[5,1],[5,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r04-linear-program-duality-0030","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-program-duality"],"title_zh":"線性規劃對偶","stem_zh":"固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-weak-duality-chain；資料={\"A\":[[1,0],[0,1]],\"b\":[6,7],\"c\":[1,5],\"x\":[[6,2],[7,2]],\"y\":[2,6]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 7/2], [2, 6], [3, 7/2], [2, 6], 是, 是, 41/2, 27, 54, 是, 是, 是, cTx_le_yTAx_le_bTy_for_every_feasible_pair)","explanation_zh":"Wave6第九層核對=([3, 7/2], [2, 6], [3, 7/2], [2, 6], 是, 是, 41/2, 27, 54, 是, 是, 是, cTx_le_yTAx_le_bTy_for_every_feasible_pair)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性規劃對偶"]},"variant":{"family_id":"qf-mod46-lpdual-weak-duality-chain","seed":202611154030,"parameters":{"A":[[1,0],[0,1]],"b":[6,7],"c":[1,5],"x":[[6,2],[7,2]],"y":[2,6],"domain":"Exact_primal_dual_linear_program","operation":"lpdual-weak-duality-chain","semantic_template":"qf-mod46-lpdual-weak-duality-chain/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[6,1],[7,1]],\"c\":[[1,1],[5,1]],\"x\":[[3,1],[7,2]],\"y\":[[2,1],[6,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r04-linear-program-duality-0031","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-program-duality"],"title_zh":"線性規劃對偶","stem_zh":"固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-weak-duality-chain；資料={\"A\":[[1,0],[0,1]],\"b\":[7,8],\"c\":[2,1],\"x\":[[7,2],[8,2]],\"y\":[3,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([7/2, 4], [3, 2], [7/2, 4], [3, 2], 是, 是, 11, 37/2, 37, 是, 是, 是, cTx_le_yTAx_le_bTy_for_every_feasible_pair)","explanation_zh":"Wave6第九層核對=([7/2, 4], [3, 2], [7/2, 4], [3, 2], 是, 是, 11, 37/2, 37, 是, 是, 是, cTx_le_yTAx_le_bTy_for_every_feasible_pair)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性規劃對偶"]},"variant":{"family_id":"qf-mod46-lpdual-weak-duality-chain","seed":202611154031,"parameters":{"A":[[1,0],[0,1]],"b":[7,8],"c":[2,1],"x":[[7,2],[8,2]],"y":[3,2],"domain":"Exact_primal_dual_linear_program","operation":"lpdual-weak-duality-chain","semantic_template":"qf-mod46-lpdual-weak-duality-chain/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[7,1],[8,1]],\"c\":[[2,1],[1,1]],\"x\":[[7,2],[4,1]],\"y\":[[3,1],[2,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r04-linear-program-duality-0032","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-program-duality"],"title_zh":"線性規劃對偶","stem_zh":"固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-weak-duality-chain；資料={\"A\":[[1,0],[0,1]],\"b\":[8,9],\"c\":[3,2],\"x\":[[8,2],[9,2]],\"y\":[4,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([4, 9/2], [4, 3], [4, 9/2], [4, 3], 是, 是, 21, 59/2, 59, 是, 是, 是, cTx_le_yTAx_le_bTy_for_every_feasible_pair)","explanation_zh":"Wave6第九層核對=([4, 9/2], [4, 3], [4, 9/2], [4, 3], 是, 是, 21, 59/2, 59, 是, 是, 是, cTx_le_yTAx_le_bTy_for_every_feasible_pair)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性規劃對偶"]},"variant":{"family_id":"qf-mod46-lpdual-weak-duality-chain","seed":202611154032,"parameters":{"A":[[1,0],[0,1]],"b":[8,9],"c":[3,2],"x":[[8,2],[9,2]],"y":[4,3],"domain":"Exact_primal_dual_linear_program","operation":"lpdual-weak-duality-chain","semantic_template":"qf-mod46-lpdual-weak-duality-chain/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[8,1],[9,1]],\"c\":[[3,1],[2,1]],\"x\":[[4,1],[9,2]],\"y\":[[4,1],[3,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r04-linear-program-duality-0033","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-program-duality"],"title_zh":"線性規劃對偶","stem_zh":"固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-weak-duality-chain；資料={\"A\":[[1,0],[0,1]],\"b\":[9,10],\"c\":[4,3],\"x\":[[9,2],[10,2]],\"y\":[5,4]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([9/2, 5], [5, 4], [9/2, 5], [5, 4], 是, 是, 33, 85/2, 85, 是, 是, 是, cTx_le_yTAx_le_bTy_for_every_feasible_pair)","explanation_zh":"Wave6第九層核對=([9/2, 5], [5, 4], [9/2, 5], [5, 4], 是, 是, 33, 85/2, 85, 是, 是, 是, cTx_le_yTAx_le_bTy_for_every_feasible_pair)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性規劃對偶"]},"variant":{"family_id":"qf-mod46-lpdual-weak-duality-chain","seed":202611154033,"parameters":{"A":[[1,0],[0,1]],"b":[9,10],"c":[4,3],"x":[[9,2],[10,2]],"y":[5,4],"domain":"Exact_primal_dual_linear_program","operation":"lpdual-weak-duality-chain","semantic_template":"qf-mod46-lpdual-weak-duality-chain/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[9,1],[10,1]],\"c\":[[4,1],[3,1]],\"x\":[[9,2],[5,1]],\"y\":[[5,1],[4,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r04-linear-program-duality-0034","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-program-duality"],"title_zh":"線性規劃對偶","stem_zh":"固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-weak-duality-chain；資料={\"A\":[[1,0],[0,1]],\"b\":[10,11],\"c\":[1,4],\"x\":[[10,2],[11,2]],\"y\":[2,5]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([5, 11/2], [2, 5], [5, 11/2], [2, 5], 是, 是, 27, 75/2, 75, 是, 是, 是, cTx_le_yTAx_le_bTy_for_every_feasible_pair)","explanation_zh":"Wave6第九層核對=([5, 11/2], [2, 5], [5, 11/2], [2, 5], 是, 是, 27, 75/2, 75, 是, 是, 是, cTx_le_yTAx_le_bTy_for_every_feasible_pair)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性規劃對偶"]},"variant":{"family_id":"qf-mod46-lpdual-weak-duality-chain","seed":202611154034,"parameters":{"A":[[1,0],[0,1]],"b":[10,11],"c":[1,4],"x":[[10,2],[11,2]],"y":[2,5],"domain":"Exact_primal_dual_linear_program","operation":"lpdual-weak-duality-chain","semantic_template":"qf-mod46-lpdual-weak-duality-chain/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[10,1],[11,1]],\"c\":[[1,1],[4,1]],\"x\":[[5,1],[11,2]],\"y\":[[2,1],[5,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r04-linear-program-duality-0035","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-program-duality"],"title_zh":"線性規劃對偶","stem_zh":"固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-weak-duality-chain；資料={\"A\":[[1,0],[0,1]],\"b\":[11,12],\"c\":[2,5],\"x\":[[11,2],[12,2]],\"y\":[3,6]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([11/2, 6], [3, 6], [11/2, 6], [3, 6], 是, 是, 41, 105/2, 105, 是, 是, 是, cTx_le_yTAx_le_bTy_for_every_feasible_pair)","explanation_zh":"Wave6第九層核對=([11/2, 6], [3, 6], [11/2, 6], [3, 6], 是, 是, 41, 105/2, 105, 是, 是, 是, cTx_le_yTAx_le_bTy_for_every_feasible_pair)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性規劃對偶"]},"variant":{"family_id":"qf-mod46-lpdual-weak-duality-chain","seed":202611154035,"parameters":{"A":[[1,0],[0,1]],"b":[11,12],"c":[2,5],"x":[[11,2],[12,2]],"y":[3,6],"domain":"Exact_primal_dual_linear_program","operation":"lpdual-weak-duality-chain","semantic_template":"qf-mod46-lpdual-weak-duality-chain/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[11,1],[12,1]],\"c\":[[2,1],[5,1]],\"x\":[[11,2],[6,1]],\"y\":[[3,1],[6,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r04-linear-program-duality-0036","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-program-duality"],"title_zh":"線性規劃對偶","stem_zh":"固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-weak-duality-chain；資料={\"A\":[[1,0],[0,1]],\"b\":[12,13],\"c\":[3,1],\"x\":[[12,2],[13,2]],\"y\":[4,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([6, 13/2], [4, 2], [6, 13/2], [4, 2], 是, 是, 49/2, 37, 74, 是, 是, 是, cTx_le_yTAx_le_bTy_for_every_feasible_pair)","explanation_zh":"Wave6第九層核對=([6, 13/2], [4, 2], [6, 13/2], [4, 2], 是, 是, 49/2, 37, 74, 是, 是, 是, cTx_le_yTAx_le_bTy_for_every_feasible_pair)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性規劃對偶"]},"variant":{"family_id":"qf-mod46-lpdual-weak-duality-chain","seed":202611154036,"parameters":{"A":[[1,0],[0,1]],"b":[12,13],"c":[3,1],"x":[[12,2],[13,2]],"y":[4,2],"domain":"Exact_primal_dual_linear_program","operation":"lpdual-weak-duality-chain","semantic_template":"qf-mod46-lpdual-weak-duality-chain/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[12,1],[13,1]],\"c\":[[3,1],[1,1]],\"x\":[[6,1],[13,2]],\"y\":[[4,1],[2,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r04-linear-program-duality-0037","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-program-duality"],"title_zh":"線性規劃對偶","stem_zh":"固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-weak-duality-chain；資料={\"A\":[[1,0],[0,1]],\"b\":[13,14],\"c\":[4,2],\"x\":[[13,2],[14,2]],\"y\":[5,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([13/2, 7], [5, 3], [13/2, 7], [5, 3], 是, 是, 40, 107/2, 107, 是, 是, 是, cTx_le_yTAx_le_bTy_for_every_feasible_pair)","explanation_zh":"Wave6第九層核對=([13/2, 7], [5, 3], [13/2, 7], [5, 3], 是, 是, 40, 107/2, 107, 是, 是, 是, cTx_le_yTAx_le_bTy_for_every_feasible_pair)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性規劃對偶"]},"variant":{"family_id":"qf-mod46-lpdual-weak-duality-chain","seed":202611154037,"parameters":{"A":[[1,0],[0,1]],"b":[13,14],"c":[4,2],"x":[[13,2],[14,2]],"y":[5,3],"domain":"Exact_primal_dual_linear_program","operation":"lpdual-weak-duality-chain","semantic_template":"qf-mod46-lpdual-weak-duality-chain/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[13,1],[14,1]],\"c\":[[4,1],[2,1]],\"x\":[[13,2],[7,1]],\"y\":[[5,1],[3,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r04-linear-program-duality-0038","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-program-duality"],"title_zh":"線性規劃對偶","stem_zh":"固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-weak-duality-chain；資料={\"A\":[[1,0],[0,1]],\"b\":[14,15],\"c\":[1,3],\"x\":[[14,2],[15,2]],\"y\":[2,4]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([7, 15/2], [2, 4], [7, 15/2], [2, 4], 是, 是, 59/2, 44, 88, 是, 是, 是, cTx_le_yTAx_le_bTy_for_every_feasible_pair)","explanation_zh":"Wave6第九層核對=([7, 15/2], [2, 4], [7, 15/2], [2, 4], 是, 是, 59/2, 44, 88, 是, 是, 是, cTx_le_yTAx_le_bTy_for_every_feasible_pair)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性規劃對偶"]},"variant":{"family_id":"qf-mod46-lpdual-weak-duality-chain","seed":202611154038,"parameters":{"A":[[1,0],[0,1]],"b":[14,15],"c":[1,3],"x":[[14,2],[15,2]],"y":[2,4],"domain":"Exact_primal_dual_linear_program","operation":"lpdual-weak-duality-chain","semantic_template":"qf-mod46-lpdual-weak-duality-chain/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[14,1],[15,1]],\"c\":[[1,1],[3,1]],\"x\":[[7,1],[15,2]],\"y\":[[2,1],[4,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r04-linear-program-duality-0039","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-program-duality"],"title_zh":"線性規劃對偶","stem_zh":"固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-weak-duality-chain；資料={\"A\":[[1,0],[0,1]],\"b\":[15,16],\"c\":[2,4],\"x\":[[15,2],[16,2]],\"y\":[3,5]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([15/2, 8], [3, 5], [15/2, 8], [3, 5], 是, 是, 47, 125/2, 125, 是, 是, 是, cTx_le_yTAx_le_bTy_for_every_feasible_pair)","explanation_zh":"Wave6第九層核對=([15/2, 8], [3, 5], [15/2, 8], [3, 5], 是, 是, 47, 125/2, 125, 是, 是, 是, cTx_le_yTAx_le_bTy_for_every_feasible_pair)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性規劃對偶"]},"variant":{"family_id":"qf-mod46-lpdual-weak-duality-chain","seed":202611154039,"parameters":{"A":[[1,0],[0,1]],"b":[15,16],"c":[2,4],"x":[[15,2],[16,2]],"y":[3,5],"domain":"Exact_primal_dual_linear_program","operation":"lpdual-weak-duality-chain","semantic_template":"qf-mod46-lpdual-weak-duality-chain/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[15,1],[16,1]],\"c\":[[2,1],[4,1]],\"x\":[[15,2],[8,1]],\"y\":[[3,1],[5,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r04-linear-program-duality-0040","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-program-duality"],"title_zh":"線性規劃對偶","stem_zh":"固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-weak-duality-chain；資料={\"A\":[[1,0],[0,1]],\"b\":[16,17],\"c\":[3,5],\"x\":[[16,2],[17,2]],\"y\":[4,6]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([8, 17/2], [4, 6], [8, 17/2], [4, 6], 是, 是, 133/2, 83, 166, 是, 是, 是, cTx_le_yTAx_le_bTy_for_every_feasible_pair)","explanation_zh":"Wave6第九層核對=([8, 17/2], [4, 6], [8, 17/2], [4, 6], 是, 是, 133/2, 83, 166, 是, 是, 是, cTx_le_yTAx_le_bTy_for_every_feasible_pair)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性規劃對偶"]},"variant":{"family_id":"qf-mod46-lpdual-weak-duality-chain","seed":202611154040,"parameters":{"A":[[1,0],[0,1]],"b":[16,17],"c":[3,5],"x":[[16,2],[17,2]],"y":[4,6],"domain":"Exact_primal_dual_linear_program","operation":"lpdual-weak-duality-chain","semantic_template":"qf-mod46-lpdual-weak-duality-chain/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[16,1],[17,1]],\"c\":[[3,1],[5,1]],\"x\":[[8,1],[17,2]],\"y\":[[4,1],[6,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r04-linear-program-duality-0041","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-program-duality"],"title_zh":"線性規劃對偶","stem_zh":"固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-weak-duality-chain；資料={\"A\":[[1,0],[0,1]],\"b\":[17,18],\"c\":[4,1],\"x\":[[17,2],[18,2]],\"y\":[5,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([17/2, 9], [5, 2], [17/2, 9], [5, 2], 是, 是, 43, 121/2, 121, 是, 是, 是, cTx_le_yTAx_le_bTy_for_every_feasible_pair)","explanation_zh":"Wave6第九層核對=([17/2, 9], [5, 2], [17/2, 9], [5, 2], 是, 是, 43, 121/2, 121, 是, 是, 是, cTx_le_yTAx_le_bTy_for_every_feasible_pair)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性規劃對偶"]},"variant":{"family_id":"qf-mod46-lpdual-weak-duality-chain","seed":202611154041,"parameters":{"A":[[1,0],[0,1]],"b":[17,18],"c":[4,1],"x":[[17,2],[18,2]],"y":[5,2],"domain":"Exact_primal_dual_linear_program","operation":"lpdual-weak-duality-chain","semantic_template":"qf-mod46-lpdual-weak-duality-chain/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[17,1],[18,1]],\"c\":[[4,1],[1,1]],\"x\":[[17,2],[9,1]],\"y\":[[5,1],[2,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r04-linear-program-duality-0042","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-program-duality"],"title_zh":"線性規劃對偶","stem_zh":"固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-weak-duality-chain；資料={\"A\":[[1,0],[0,1]],\"b\":[18,19],\"c\":[1,2],\"x\":[[18,2],[19,2]],\"y\":[2,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([9, 19/2], [2, 3], [9, 19/2], [2, 3], 是, 是, 28, 93/2, 93, 是, 是, 是, cTx_le_yTAx_le_bTy_for_every_feasible_pair)","explanation_zh":"Wave6第九層核對=([9, 19/2], [2, 3], [9, 19/2], [2, 3], 是, 是, 28, 93/2, 93, 是, 是, 是, cTx_le_yTAx_le_bTy_for_every_feasible_pair)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性規劃對偶"]},"variant":{"family_id":"qf-mod46-lpdual-weak-duality-chain","seed":202611154042,"parameters":{"A":[[1,0],[0,1]],"b":[18,19],"c":[1,2],"x":[[18,2],[19,2]],"y":[2,3],"domain":"Exact_primal_dual_linear_program","operation":"lpdual-weak-duality-chain","semantic_template":"qf-mod46-lpdual-weak-duality-chain/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[18,1],[19,1]],\"c\":[[1,1],[2,1]],\"x\":[[9,1],[19,2]],\"y\":[[2,1],[3,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r04-linear-program-duality-0043","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-program-duality"],"title_zh":"線性規劃對偶","stem_zh":"固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-weak-duality-chain；資料={\"A\":[[1,0],[0,1]],\"b\":[19,20],\"c\":[2,3],\"x\":[[19,2],[20,2]],\"y\":[3,4]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([19/2, 10], [3, 4], [19/2, 10], [3, 4], 是, 是, 49, 137/2, 137, 是, 是, 是, cTx_le_yTAx_le_bTy_for_every_feasible_pair)","explanation_zh":"Wave6第九層核對=([19/2, 10], [3, 4], [19/2, 10], [3, 4], 是, 是, 49, 137/2, 137, 是, 是, 是, cTx_le_yTAx_le_bTy_for_every_feasible_pair)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性規劃對偶"]},"variant":{"family_id":"qf-mod46-lpdual-weak-duality-chain","seed":202611154043,"parameters":{"A":[[1,0],[0,1]],"b":[19,20],"c":[2,3],"x":[[19,2],[20,2]],"y":[3,4],"domain":"Exact_primal_dual_linear_program","operation":"lpdual-weak-duality-chain","semantic_template":"qf-mod46-lpdual-weak-duality-chain/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[19,1],[20,1]],\"c\":[[2,1],[3,1]],\"x\":[[19,2],[10,1]],\"y\":[[3,1],[4,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r04-linear-program-duality-0044","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-program-duality"],"title_zh":"線性規劃對偶","stem_zh":"固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-weak-duality-chain；資料={\"A\":[[1,0],[0,1]],\"b\":[20,21],\"c\":[3,4],\"x\":[[20,2],[21,2]],\"y\":[4,5]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([10, 21/2], [4, 5], [10, 21/2], [4, 5], 是, 是, 72, 185/2, 185, 是, 是, 是, cTx_le_yTAx_le_bTy_for_every_feasible_pair)","explanation_zh":"Wave6第九層核對=([10, 21/2], [4, 5], [10, 21/2], [4, 5], 是, 是, 72, 185/2, 185, 是, 是, 是, cTx_le_yTAx_le_bTy_for_every_feasible_pair)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性規劃對偶"]},"variant":{"family_id":"qf-mod46-lpdual-weak-duality-chain","seed":202611154044,"parameters":{"A":[[1,0],[0,1]],"b":[20,21],"c":[3,4],"x":[[20,2],[21,2]],"y":[4,5],"domain":"Exact_primal_dual_linear_program","operation":"lpdual-weak-duality-chain","semantic_template":"qf-mod46-lpdual-weak-duality-chain/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[20,1],[21,1]],\"c\":[[3,1],[4,1]],\"x\":[[10,1],[21,2]],\"y\":[[4,1],[5,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r04-linear-program-duality-0045","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-program-duality"],"title_zh":"線性規劃對偶","stem_zh":"固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-weak-duality-chain；資料={\"A\":[[1,0],[0,1]],\"b\":[21,22],\"c\":[4,5],\"x\":[[21,2],[22,2]],\"y\":[5,6]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([21/2, 11], [5, 6], [21/2, 11], [5, 6], 是, 是, 97, 237/2, 237, 是, 是, 是, cTx_le_yTAx_le_bTy_for_every_feasible_pair)","explanation_zh":"Wave6第九層核對=([21/2, 11], [5, 6], [21/2, 11], [5, 6], 是, 是, 97, 237/2, 237, 是, 是, 是, cTx_le_yTAx_le_bTy_for_every_feasible_pair)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性規劃對偶"]},"variant":{"family_id":"qf-mod46-lpdual-weak-duality-chain","seed":202611154045,"parameters":{"A":[[1,0],[0,1]],"b":[21,22],"c":[4,5],"x":[[21,2],[22,2]],"y":[5,6],"domain":"Exact_primal_dual_linear_program","operation":"lpdual-weak-duality-chain","semantic_template":"qf-mod46-lpdual-weak-duality-chain/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[21,1],[22,1]],\"c\":[[4,1],[5,1]],\"x\":[[21,2],[11,1]],\"y\":[[5,1],[6,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r04-linear-program-duality-0046","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-program-duality"],"title_zh":"線性規劃對偶","stem_zh":"固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-weak-duality-chain；資料={\"A\":[[1,0],[0,1]],\"b\":[22,23],\"c\":[1,1],\"x\":[[22,2],[23,2]],\"y\":[2,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([11, 23/2], [2, 2], [11, 23/2], [2, 2], 是, 是, 45/2, 45, 90, 是, 是, 是, cTx_le_yTAx_le_bTy_for_every_feasible_pair)","explanation_zh":"Wave6第九層核對=([11, 23/2], [2, 2], [11, 23/2], [2, 2], 是, 是, 45/2, 45, 90, 是, 是, 是, cTx_le_yTAx_le_bTy_for_every_feasible_pair)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性規劃對偶"]},"variant":{"family_id":"qf-mod46-lpdual-weak-duality-chain","seed":202611154046,"parameters":{"A":[[1,0],[0,1]],"b":[22,23],"c":[1,1],"x":[[22,2],[23,2]],"y":[2,2],"domain":"Exact_primal_dual_linear_program","operation":"lpdual-weak-duality-chain","semantic_template":"qf-mod46-lpdual-weak-duality-chain/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[22,1],[23,1]],\"c\":[[1,1],[1,1]],\"x\":[[11,1],[23,2]],\"y\":[[2,1],[2,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r04-linear-program-duality-0047","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-program-duality"],"title_zh":"線性規劃對偶","stem_zh":"固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-weak-duality-chain；資料={\"A\":[[1,0],[0,1]],\"b\":[23,24],\"c\":[2,2],\"x\":[[23,2],[24,2]],\"y\":[3,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([23/2, 12], [3, 3], [23/2, 12], [3, 3], 是, 是, 47, 141/2, 141, 是, 是, 是, cTx_le_yTAx_le_bTy_for_every_feasible_pair)","explanation_zh":"Wave6第九層核對=([23/2, 12], [3, 3], [23/2, 12], [3, 3], 是, 是, 47, 141/2, 141, 是, 是, 是, cTx_le_yTAx_le_bTy_for_every_feasible_pair)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性規劃對偶"]},"variant":{"family_id":"qf-mod46-lpdual-weak-duality-chain","seed":202611154047,"parameters":{"A":[[1,0],[0,1]],"b":[23,24],"c":[2,2],"x":[[23,2],[24,2]],"y":[3,3],"domain":"Exact_primal_dual_linear_program","operation":"lpdual-weak-duality-chain","semantic_template":"qf-mod46-lpdual-weak-duality-chain/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[23,1],[24,1]],\"c\":[[2,1],[2,1]],\"x\":[[23,2],[12,1]],\"y\":[[3,1],[3,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r04-linear-program-duality-0048","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-program-duality"],"title_zh":"線性規劃對偶","stem_zh":"固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-weak-duality-chain；資料={\"A\":[[1,0],[0,1]],\"b\":[24,25],\"c\":[3,3],\"x\":[[24,2],[25,2]],\"y\":[4,4]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([12, 25/2], [4, 4], [12, 25/2], [4, 4], 是, 是, 147/2, 98, 196, 是, 是, 是, cTx_le_yTAx_le_bTy_for_every_feasible_pair)","explanation_zh":"Wave6第九層核對=([12, 25/2], [4, 4], [12, 25/2], [4, 4], 是, 是, 147/2, 98, 196, 是, 是, 是, cTx_le_yTAx_le_bTy_for_every_feasible_pair)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性規劃對偶"]},"variant":{"family_id":"qf-mod46-lpdual-weak-duality-chain","seed":202611154048,"parameters":{"A":[[1,0],[0,1]],"b":[24,25],"c":[3,3],"x":[[24,2],[25,2]],"y":[4,4],"domain":"Exact_primal_dual_linear_program","operation":"lpdual-weak-duality-chain","semantic_template":"qf-mod46-lpdual-weak-duality-chain/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[24,1],[25,1]],\"c\":[[3,1],[3,1]],\"x\":[[12,1],[25,2]],\"y\":[[4,1],[4,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r04-linear-program-duality-0049","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-program-duality"],"title_zh":"線性規劃對偶","stem_zh":"固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-weak-duality-chain；資料={\"A\":[[1,0],[0,1]],\"b\":[25,26],\"c\":[4,4],\"x\":[[25,2],[26,2]],\"y\":[5,5]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([25/2, 13], [5, 5], [25/2, 13], [5, 5], 是, 是, 102, 255/2, 255, 是, 是, 是, cTx_le_yTAx_le_bTy_for_every_feasible_pair)","explanation_zh":"Wave6第九層核對=([25/2, 13], [5, 5], [25/2, 13], [5, 5], 是, 是, 102, 255/2, 255, 是, 是, 是, cTx_le_yTAx_le_bTy_for_every_feasible_pair)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性規劃對偶"]},"variant":{"family_id":"qf-mod46-lpdual-weak-duality-chain","seed":202611154049,"parameters":{"A":[[1,0],[0,1]],"b":[25,26],"c":[4,4],"x":[[25,2],[26,2]],"y":[5,5],"domain":"Exact_primal_dual_linear_program","operation":"lpdual-weak-duality-chain","semantic_template":"qf-mod46-lpdual-weak-duality-chain/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[25,1],[26,1]],\"c\":[[4,1],[4,1]],\"x\":[[25,2],[13,1]],\"y\":[[5,1],[5,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r04-linear-program-duality-0050","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-program-duality"],"title_zh":"線性規劃對偶","stem_zh":"固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-weak-duality-chain；資料={\"A\":[[1,0],[0,1]],\"b\":[26,27],\"c\":[1,5],\"x\":[[26,2],[27,2]],\"y\":[2,6]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([13, 27/2], [2, 6], [13, 27/2], [2, 6], 是, 是, 161/2, 107, 214, 是, 是, 是, cTx_le_yTAx_le_bTy_for_every_feasible_pair)","explanation_zh":"Wave6第九層核對=([13, 27/2], [2, 6], [13, 27/2], [2, 6], 是, 是, 161/2, 107, 214, 是, 是, 是, cTx_le_yTAx_le_bTy_for_every_feasible_pair)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性規劃對偶"]},"variant":{"family_id":"qf-mod46-lpdual-weak-duality-chain","seed":202611154050,"parameters":{"A":[[1,0],[0,1]],"b":[26,27],"c":[1,5],"x":[[26,2],[27,2]],"y":[2,6],"domain":"Exact_primal_dual_linear_program","operation":"lpdual-weak-duality-chain","semantic_template":"qf-mod46-lpdual-weak-duality-chain/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[26,1],[27,1]],\"c\":[[1,1],[5,1]],\"x\":[[13,1],[27,2]],\"y\":[[2,1],[6,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r04-linear-program-duality-0051","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-program-duality"],"title_zh":"線性規劃對偶","stem_zh":"固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-zero-gap-certificate；資料={\"A\":[[1,0],[0,1]],\"b\":[2,3],\"c\":[1,1],\"x\":[2,3],\"y\":[1,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([2, 3], [1, 1], [2, 3], [1, 1], 是, 是, 5, 5, 5, 是, 是, 是, 是, 是, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)","explanation_zh":"Wave6第九層核對=([2, 3], [1, 1], [2, 3], [1, 1], 是, 是, 5, 5, 5, 是, 是, 是, 是, 是, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性規劃對偶"]},"variant":{"family_id":"qf-mod46-lpdual-zero-gap-certificate","seed":202611154051,"parameters":{"A":[[1,0],[0,1]],"b":[2,3],"c":[1,1],"x":[2,3],"y":[1,1],"domain":"Exact_primal_dual_linear_program","operation":"lpdual-zero-gap-certificate","semantic_template":"qf-mod46-lpdual-zero-gap-certificate/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[2,1],[3,1]],\"c\":[[1,1],[1,1]],\"x\":[[2,1],[3,1]],\"y\":[[1,1],[1,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r04-linear-program-duality-0052","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-program-duality"],"title_zh":"線性規劃對偶","stem_zh":"固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-zero-gap-certificate；資料={\"A\":[[1,0],[0,1]],\"b\":[3,4],\"c\":[2,2],\"x\":[3,4],\"y\":[2,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([3, 4], [2, 2], [3, 4], [2, 2], 是, 是, 14, 14, 14, 是, 是, 是, 是, 是, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)","explanation_zh":"Wave6第九層核對=([3, 4], [2, 2], [3, 4], [2, 2], 是, 是, 14, 14, 14, 是, 是, 是, 是, 是, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性規劃對偶"]},"variant":{"family_id":"qf-mod46-lpdual-zero-gap-certificate","seed":202611154052,"parameters":{"A":[[1,0],[0,1]],"b":[3,4],"c":[2,2],"x":[3,4],"y":[2,2],"domain":"Exact_primal_dual_linear_program","operation":"lpdual-zero-gap-certificate","semantic_template":"qf-mod46-lpdual-zero-gap-certificate/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[3,1],[4,1]],\"c\":[[2,1],[2,1]],\"x\":[[3,1],[4,1]],\"y\":[[2,1],[2,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r04-linear-program-duality-0053","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-program-duality"],"title_zh":"線性規劃對偶","stem_zh":"固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-zero-gap-certificate；資料={\"A\":[[1,0],[0,1]],\"b\":[4,5],\"c\":[3,3],\"x\":[4,5],\"y\":[3,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([4, 5], [3, 3], [4, 5], [3, 3], 是, 是, 27, 27, 27, 是, 是, 是, 是, 是, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)","explanation_zh":"Wave6第九層核對=([4, 5], [3, 3], [4, 5], [3, 3], 是, 是, 27, 27, 27, 是, 是, 是, 是, 是, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性規劃對偶"]},"variant":{"family_id":"qf-mod46-lpdual-zero-gap-certificate","seed":202611154053,"parameters":{"A":[[1,0],[0,1]],"b":[4,5],"c":[3,3],"x":[4,5],"y":[3,3],"domain":"Exact_primal_dual_linear_program","operation":"lpdual-zero-gap-certificate","semantic_template":"qf-mod46-lpdual-zero-gap-certificate/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[4,1],[5,1]],\"c\":[[3,1],[3,1]],\"x\":[[4,1],[5,1]],\"y\":[[3,1],[3,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r04-linear-program-duality-0054","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-program-duality"],"title_zh":"線性規劃對偶","stem_zh":"固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-zero-gap-certificate；資料={\"A\":[[1,0],[0,1]],\"b\":[5,6],\"c\":[4,4],\"x\":[5,6],\"y\":[4,4]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([5, 6], [4, 4], [5, 6], [4, 4], 是, 是, 44, 44, 44, 是, 是, 是, 是, 是, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)","explanation_zh":"Wave6第九層核對=([5, 6], [4, 4], [5, 6], [4, 4], 是, 是, 44, 44, 44, 是, 是, 是, 是, 是, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性規劃對偶"]},"variant":{"family_id":"qf-mod46-lpdual-zero-gap-certificate","seed":202611154054,"parameters":{"A":[[1,0],[0,1]],"b":[5,6],"c":[4,4],"x":[5,6],"y":[4,4],"domain":"Exact_primal_dual_linear_program","operation":"lpdual-zero-gap-certificate","semantic_template":"qf-mod46-lpdual-zero-gap-certificate/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[5,1],[6,1]],\"c\":[[4,1],[4,1]],\"x\":[[5,1],[6,1]],\"y\":[[4,1],[4,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r04-linear-program-duality-0055","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-program-duality"],"title_zh":"線性規劃對偶","stem_zh":"固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-zero-gap-certificate；資料={\"A\":[[1,0],[0,1]],\"b\":[6,7],\"c\":[1,5],\"x\":[6,7],\"y\":[1,5]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([6, 7], [1, 5], [6, 7], [1, 5], 是, 是, 41, 41, 41, 是, 是, 是, 是, 是, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)","explanation_zh":"Wave6第九層核對=([6, 7], [1, 5], [6, 7], [1, 5], 是, 是, 41, 41, 41, 是, 是, 是, 是, 是, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性規劃對偶"]},"variant":{"family_id":"qf-mod46-lpdual-zero-gap-certificate","seed":202611154055,"parameters":{"A":[[1,0],[0,1]],"b":[6,7],"c":[1,5],"x":[6,7],"y":[1,5],"domain":"Exact_primal_dual_linear_program","operation":"lpdual-zero-gap-certificate","semantic_template":"qf-mod46-lpdual-zero-gap-certificate/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[6,1],[7,1]],\"c\":[[1,1],[5,1]],\"x\":[[6,1],[7,1]],\"y\":[[1,1],[5,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r04-linear-program-duality-0056","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-program-duality"],"title_zh":"線性規劃對偶","stem_zh":"固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-zero-gap-certificate；資料={\"A\":[[1,0],[0,1]],\"b\":[7,8],\"c\":[2,1],\"x\":[7,8],\"y\":[2,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([7, 8], [2, 1], [7, 8], [2, 1], 是, 是, 22, 22, 22, 是, 是, 是, 是, 是, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)","explanation_zh":"Wave6第九層核對=([7, 8], [2, 1], [7, 8], [2, 1], 是, 是, 22, 22, 22, 是, 是, 是, 是, 是, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性規劃對偶"]},"variant":{"family_id":"qf-mod46-lpdual-zero-gap-certificate","seed":202611154056,"parameters":{"A":[[1,0],[0,1]],"b":[7,8],"c":[2,1],"x":[7,8],"y":[2,1],"domain":"Exact_primal_dual_linear_program","operation":"lpdual-zero-gap-certificate","semantic_template":"qf-mod46-lpdual-zero-gap-certificate/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[7,1],[8,1]],\"c\":[[2,1],[1,1]],\"x\":[[7,1],[8,1]],\"y\":[[2,1],[1,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r04-linear-program-duality-0057","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-program-duality"],"title_zh":"線性規劃對偶","stem_zh":"固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-zero-gap-certificate；資料={\"A\":[[1,0],[0,1]],\"b\":[8,9],\"c\":[3,2],\"x\":[8,9],\"y\":[3,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([8, 9], [3, 2], [8, 9], [3, 2], 是, 是, 42, 42, 42, 是, 是, 是, 是, 是, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)","explanation_zh":"Wave6第九層核對=([8, 9], [3, 2], [8, 9], [3, 2], 是, 是, 42, 42, 42, 是, 是, 是, 是, 是, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性規劃對偶"]},"variant":{"family_id":"qf-mod46-lpdual-zero-gap-certificate","seed":202611154057,"parameters":{"A":[[1,0],[0,1]],"b":[8,9],"c":[3,2],"x":[8,9],"y":[3,2],"domain":"Exact_primal_dual_linear_program","operation":"lpdual-zero-gap-certificate","semantic_template":"qf-mod46-lpdual-zero-gap-certificate/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[8,1],[9,1]],\"c\":[[3,1],[2,1]],\"x\":[[8,1],[9,1]],\"y\":[[3,1],[2,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r04-linear-program-duality-0058","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-program-duality"],"title_zh":"線性規劃對偶","stem_zh":"固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-zero-gap-certificate；資料={\"A\":[[1,0],[0,1]],\"b\":[9,10],\"c\":[4,3],\"x\":[9,10],\"y\":[4,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([9, 10], [4, 3], [9, 10], [4, 3], 是, 是, 66, 66, 66, 是, 是, 是, 是, 是, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)","explanation_zh":"Wave6第九層核對=([9, 10], [4, 3], [9, 10], [4, 3], 是, 是, 66, 66, 66, 是, 是, 是, 是, 是, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性規劃對偶"]},"variant":{"family_id":"qf-mod46-lpdual-zero-gap-certificate","seed":202611154058,"parameters":{"A":[[1,0],[0,1]],"b":[9,10],"c":[4,3],"x":[9,10],"y":[4,3],"domain":"Exact_primal_dual_linear_program","operation":"lpdual-zero-gap-certificate","semantic_template":"qf-mod46-lpdual-zero-gap-certificate/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[9,1],[10,1]],\"c\":[[4,1],[3,1]],\"x\":[[9,1],[10,1]],\"y\":[[4,1],[3,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r04-linear-program-duality-0059","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-program-duality"],"title_zh":"線性規劃對偶","stem_zh":"固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-zero-gap-certificate；資料={\"A\":[[1,0],[0,1]],\"b\":[10,11],\"c\":[1,4],\"x\":[10,11],\"y\":[1,4]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([10, 11], [1, 4], [10, 11], [1, 4], 是, 是, 54, 54, 54, 是, 是, 是, 是, 是, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)","explanation_zh":"Wave6第九層核對=([10, 11], [1, 4], [10, 11], [1, 4], 是, 是, 54, 54, 54, 是, 是, 是, 是, 是, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性規劃對偶"]},"variant":{"family_id":"qf-mod46-lpdual-zero-gap-certificate","seed":202611154059,"parameters":{"A":[[1,0],[0,1]],"b":[10,11],"c":[1,4],"x":[10,11],"y":[1,4],"domain":"Exact_primal_dual_linear_program","operation":"lpdual-zero-gap-certificate","semantic_template":"qf-mod46-lpdual-zero-gap-certificate/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[10,1],[11,1]],\"c\":[[1,1],[4,1]],\"x\":[[10,1],[11,1]],\"y\":[[1,1],[4,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r04-linear-program-duality-0060","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-program-duality"],"title_zh":"線性規劃對偶","stem_zh":"固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-zero-gap-certificate；資料={\"A\":[[1,0],[0,1]],\"b\":[11,12],\"c\":[2,5],\"x\":[11,12],\"y\":[2,5]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([11, 12], [2, 5], [11, 12], [2, 5], 是, 是, 82, 82, 82, 是, 是, 是, 是, 是, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)","explanation_zh":"Wave6第九層核對=([11, 12], [2, 5], [11, 12], [2, 5], 是, 是, 82, 82, 82, 是, 是, 是, 是, 是, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性規劃對偶"]},"variant":{"family_id":"qf-mod46-lpdual-zero-gap-certificate","seed":202611154060,"parameters":{"A":[[1,0],[0,1]],"b":[11,12],"c":[2,5],"x":[11,12],"y":[2,5],"domain":"Exact_primal_dual_linear_program","operation":"lpdual-zero-gap-certificate","semantic_template":"qf-mod46-lpdual-zero-gap-certificate/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[11,1],[12,1]],\"c\":[[2,1],[5,1]],\"x\":[[11,1],[12,1]],\"y\":[[2,1],[5,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r04-linear-program-duality-0061","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-program-duality"],"title_zh":"線性規劃對偶","stem_zh":"固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-zero-gap-certificate；資料={\"A\":[[1,0],[0,1]],\"b\":[12,13],\"c\":[3,1],\"x\":[12,13],\"y\":[3,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([12, 13], [3, 1], [12, 13], [3, 1], 是, 是, 49, 49, 49, 是, 是, 是, 是, 是, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)","explanation_zh":"Wave6第九層核對=([12, 13], [3, 1], [12, 13], [3, 1], 是, 是, 49, 49, 49, 是, 是, 是, 是, 是, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性規劃對偶"]},"variant":{"family_id":"qf-mod46-lpdual-zero-gap-certificate","seed":202611154061,"parameters":{"A":[[1,0],[0,1]],"b":[12,13],"c":[3,1],"x":[12,13],"y":[3,1],"domain":"Exact_primal_dual_linear_program","operation":"lpdual-zero-gap-certificate","semantic_template":"qf-mod46-lpdual-zero-gap-certificate/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[12,1],[13,1]],\"c\":[[3,1],[1,1]],\"x\":[[12,1],[13,1]],\"y\":[[3,1],[1,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r04-linear-program-duality-0062","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-program-duality"],"title_zh":"線性規劃對偶","stem_zh":"固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-zero-gap-certificate；資料={\"A\":[[1,0],[0,1]],\"b\":[13,14],\"c\":[4,2],\"x\":[13,14],\"y\":[4,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([13, 14], [4, 2], [13, 14], [4, 2], 是, 是, 80, 80, 80, 是, 是, 是, 是, 是, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)","explanation_zh":"Wave6第九層核對=([13, 14], [4, 2], [13, 14], [4, 2], 是, 是, 80, 80, 80, 是, 是, 是, 是, 是, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性規劃對偶"]},"variant":{"family_id":"qf-mod46-lpdual-zero-gap-certificate","seed":202611154062,"parameters":{"A":[[1,0],[0,1]],"b":[13,14],"c":[4,2],"x":[13,14],"y":[4,2],"domain":"Exact_primal_dual_linear_program","operation":"lpdual-zero-gap-certificate","semantic_template":"qf-mod46-lpdual-zero-gap-certificate/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[13,1],[14,1]],\"c\":[[4,1],[2,1]],\"x\":[[13,1],[14,1]],\"y\":[[4,1],[2,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r04-linear-program-duality-0063","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-program-duality"],"title_zh":"線性規劃對偶","stem_zh":"固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-zero-gap-certificate；資料={\"A\":[[1,0],[0,1]],\"b\":[14,15],\"c\":[1,3],\"x\":[14,15],\"y\":[1,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([14, 15], [1, 3], [14, 15], [1, 3], 是, 是, 59, 59, 59, 是, 是, 是, 是, 是, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)","explanation_zh":"Wave6第九層核對=([14, 15], [1, 3], [14, 15], [1, 3], 是, 是, 59, 59, 59, 是, 是, 是, 是, 是, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性規劃對偶"]},"variant":{"family_id":"qf-mod46-lpdual-zero-gap-certificate","seed":202611154063,"parameters":{"A":[[1,0],[0,1]],"b":[14,15],"c":[1,3],"x":[14,15],"y":[1,3],"domain":"Exact_primal_dual_linear_program","operation":"lpdual-zero-gap-certificate","semantic_template":"qf-mod46-lpdual-zero-gap-certificate/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[14,1],[15,1]],\"c\":[[1,1],[3,1]],\"x\":[[14,1],[15,1]],\"y\":[[1,1],[3,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r04-linear-program-duality-0064","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-program-duality"],"title_zh":"線性規劃對偶","stem_zh":"固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-zero-gap-certificate；資料={\"A\":[[1,0],[0,1]],\"b\":[15,16],\"c\":[2,4],\"x\":[15,16],\"y\":[2,4]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([15, 16], [2, 4], [15, 16], [2, 4], 是, 是, 94, 94, 94, 是, 是, 是, 是, 是, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)","explanation_zh":"Wave6第九層核對=([15, 16], [2, 4], [15, 16], [2, 4], 是, 是, 94, 94, 94, 是, 是, 是, 是, 是, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性規劃對偶"]},"variant":{"family_id":"qf-mod46-lpdual-zero-gap-certificate","seed":202611154064,"parameters":{"A":[[1,0],[0,1]],"b":[15,16],"c":[2,4],"x":[15,16],"y":[2,4],"domain":"Exact_primal_dual_linear_program","operation":"lpdual-zero-gap-certificate","semantic_template":"qf-mod46-lpdual-zero-gap-certificate/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[15,1],[16,1]],\"c\":[[2,1],[4,1]],\"x\":[[15,1],[16,1]],\"y\":[[2,1],[4,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r04-linear-program-duality-0065","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-program-duality"],"title_zh":"線性規劃對偶","stem_zh":"固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-zero-gap-certificate；資料={\"A\":[[1,0],[0,1]],\"b\":[16,17],\"c\":[3,5],\"x\":[16,17],\"y\":[3,5]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([16, 17], [3, 5], [16, 17], [3, 5], 是, 是, 133, 133, 133, 是, 是, 是, 是, 是, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)","explanation_zh":"Wave6第九層核對=([16, 17], [3, 5], [16, 17], [3, 5], 是, 是, 133, 133, 133, 是, 是, 是, 是, 是, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性規劃對偶"]},"variant":{"family_id":"qf-mod46-lpdual-zero-gap-certificate","seed":202611154065,"parameters":{"A":[[1,0],[0,1]],"b":[16,17],"c":[3,5],"x":[16,17],"y":[3,5],"domain":"Exact_primal_dual_linear_program","operation":"lpdual-zero-gap-certificate","semantic_template":"qf-mod46-lpdual-zero-gap-certificate/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[16,1],[17,1]],\"c\":[[3,1],[5,1]],\"x\":[[16,1],[17,1]],\"y\":[[3,1],[5,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r04-linear-program-duality-0066","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-program-duality"],"title_zh":"線性規劃對偶","stem_zh":"固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-zero-gap-certificate；資料={\"A\":[[1,0],[0,1]],\"b\":[17,18],\"c\":[4,1],\"x\":[17,18],\"y\":[4,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([17, 18], [4, 1], [17, 18], [4, 1], 是, 是, 86, 86, 86, 是, 是, 是, 是, 是, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)","explanation_zh":"Wave6第九層核對=([17, 18], [4, 1], [17, 18], [4, 1], 是, 是, 86, 86, 86, 是, 是, 是, 是, 是, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性規劃對偶"]},"variant":{"family_id":"qf-mod46-lpdual-zero-gap-certificate","seed":202611154066,"parameters":{"A":[[1,0],[0,1]],"b":[17,18],"c":[4,1],"x":[17,18],"y":[4,1],"domain":"Exact_primal_dual_linear_program","operation":"lpdual-zero-gap-certificate","semantic_template":"qf-mod46-lpdual-zero-gap-certificate/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[17,1],[18,1]],\"c\":[[4,1],[1,1]],\"x\":[[17,1],[18,1]],\"y\":[[4,1],[1,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r04-linear-program-duality-0067","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-program-duality"],"title_zh":"線性規劃對偶","stem_zh":"固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-zero-gap-certificate；資料={\"A\":[[1,0],[0,1]],\"b\":[18,19],\"c\":[1,2],\"x\":[18,19],\"y\":[1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([18, 19], [1, 2], [18, 19], [1, 2], 是, 是, 56, 56, 56, 是, 是, 是, 是, 是, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)","explanation_zh":"Wave6第九層核對=([18, 19], [1, 2], [18, 19], [1, 2], 是, 是, 56, 56, 56, 是, 是, 是, 是, 是, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性規劃對偶"]},"variant":{"family_id":"qf-mod46-lpdual-zero-gap-certificate","seed":202611154067,"parameters":{"A":[[1,0],[0,1]],"b":[18,19],"c":[1,2],"x":[18,19],"y":[1,2],"domain":"Exact_primal_dual_linear_program","operation":"lpdual-zero-gap-certificate","semantic_template":"qf-mod46-lpdual-zero-gap-certificate/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[18,1],[19,1]],\"c\":[[1,1],[2,1]],\"x\":[[18,1],[19,1]],\"y\":[[1,1],[2,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r04-linear-program-duality-0068","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-program-duality"],"title_zh":"線性規劃對偶","stem_zh":"固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-zero-gap-certificate；資料={\"A\":[[1,0],[0,1]],\"b\":[19,20],\"c\":[2,3],\"x\":[19,20],\"y\":[2,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([19, 20], [2, 3], [19, 20], [2, 3], 是, 是, 98, 98, 98, 是, 是, 是, 是, 是, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)","explanation_zh":"Wave6第九層核對=([19, 20], [2, 3], [19, 20], [2, 3], 是, 是, 98, 98, 98, 是, 是, 是, 是, 是, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性規劃對偶"]},"variant":{"family_id":"qf-mod46-lpdual-zero-gap-certificate","seed":202611154068,"parameters":{"A":[[1,0],[0,1]],"b":[19,20],"c":[2,3],"x":[19,20],"y":[2,3],"domain":"Exact_primal_dual_linear_program","operation":"lpdual-zero-gap-certificate","semantic_template":"qf-mod46-lpdual-zero-gap-certificate/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[19,1],[20,1]],\"c\":[[2,1],[3,1]],\"x\":[[19,1],[20,1]],\"y\":[[2,1],[3,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r04-linear-program-duality-0069","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-program-duality"],"title_zh":"線性規劃對偶","stem_zh":"固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-zero-gap-certificate；資料={\"A\":[[1,0],[0,1]],\"b\":[20,21],\"c\":[3,4],\"x\":[20,21],\"y\":[3,4]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([20, 21], [3, 4], [20, 21], [3, 4], 是, 是, 144, 144, 144, 是, 是, 是, 是, 是, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)","explanation_zh":"Wave6第九層核對=([20, 21], [3, 4], [20, 21], [3, 4], 是, 是, 144, 144, 144, 是, 是, 是, 是, 是, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性規劃對偶"]},"variant":{"family_id":"qf-mod46-lpdual-zero-gap-certificate","seed":202611154069,"parameters":{"A":[[1,0],[0,1]],"b":[20,21],"c":[3,4],"x":[20,21],"y":[3,4],"domain":"Exact_primal_dual_linear_program","operation":"lpdual-zero-gap-certificate","semantic_template":"qf-mod46-lpdual-zero-gap-certificate/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[20,1],[21,1]],\"c\":[[3,1],[4,1]],\"x\":[[20,1],[21,1]],\"y\":[[3,1],[4,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r04-linear-program-duality-0070","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-program-duality"],"title_zh":"線性規劃對偶","stem_zh":"固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-zero-gap-certificate；資料={\"A\":[[1,0],[0,1]],\"b\":[21,22],\"c\":[4,5],\"x\":[21,22],\"y\":[4,5]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([21, 22], [4, 5], [21, 22], [4, 5], 是, 是, 194, 194, 194, 是, 是, 是, 是, 是, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)","explanation_zh":"Wave6第九層核對=([21, 22], [4, 5], [21, 22], [4, 5], 是, 是, 194, 194, 194, 是, 是, 是, 是, 是, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性規劃對偶"]},"variant":{"family_id":"qf-mod46-lpdual-zero-gap-certificate","seed":202611154070,"parameters":{"A":[[1,0],[0,1]],"b":[21,22],"c":[4,5],"x":[21,22],"y":[4,5],"domain":"Exact_primal_dual_linear_program","operation":"lpdual-zero-gap-certificate","semantic_template":"qf-mod46-lpdual-zero-gap-certificate/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[21,1],[22,1]],\"c\":[[4,1],[5,1]],\"x\":[[21,1],[22,1]],\"y\":[[4,1],[5,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r04-linear-program-duality-0071","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-program-duality"],"title_zh":"線性規劃對偶","stem_zh":"固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-zero-gap-certificate；資料={\"A\":[[1,0],[0,1]],\"b\":[22,23],\"c\":[1,1],\"x\":[22,23],\"y\":[1,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([22, 23], [1, 1], [22, 23], [1, 1], 是, 是, 45, 45, 45, 是, 是, 是, 是, 是, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)","explanation_zh":"Wave6第九層核對=([22, 23], [1, 1], [22, 23], [1, 1], 是, 是, 45, 45, 45, 是, 是, 是, 是, 是, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性規劃對偶"]},"variant":{"family_id":"qf-mod46-lpdual-zero-gap-certificate","seed":202611154071,"parameters":{"A":[[1,0],[0,1]],"b":[22,23],"c":[1,1],"x":[22,23],"y":[1,1],"domain":"Exact_primal_dual_linear_program","operation":"lpdual-zero-gap-certificate","semantic_template":"qf-mod46-lpdual-zero-gap-certificate/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[22,1],[23,1]],\"c\":[[1,1],[1,1]],\"x\":[[22,1],[23,1]],\"y\":[[1,1],[1,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r04-linear-program-duality-0072","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-program-duality"],"title_zh":"線性規劃對偶","stem_zh":"固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-zero-gap-certificate；資料={\"A\":[[1,0],[0,1]],\"b\":[23,24],\"c\":[2,2],\"x\":[23,24],\"y\":[2,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([23, 24], [2, 2], [23, 24], [2, 2], 是, 是, 94, 94, 94, 是, 是, 是, 是, 是, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)","explanation_zh":"Wave6第九層核對=([23, 24], [2, 2], [23, 24], [2, 2], 是, 是, 94, 94, 94, 是, 是, 是, 是, 是, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性規劃對偶"]},"variant":{"family_id":"qf-mod46-lpdual-zero-gap-certificate","seed":202611154072,"parameters":{"A":[[1,0],[0,1]],"b":[23,24],"c":[2,2],"x":[23,24],"y":[2,2],"domain":"Exact_primal_dual_linear_program","operation":"lpdual-zero-gap-certificate","semantic_template":"qf-mod46-lpdual-zero-gap-certificate/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[23,1],[24,1]],\"c\":[[2,1],[2,1]],\"x\":[[23,1],[24,1]],\"y\":[[2,1],[2,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r04-linear-program-duality-0073","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-program-duality"],"title_zh":"線性規劃對偶","stem_zh":"固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-zero-gap-certificate；資料={\"A\":[[1,0],[0,1]],\"b\":[24,25],\"c\":[3,3],\"x\":[24,25],\"y\":[3,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([24, 25], [3, 3], [24, 25], [3, 3], 是, 是, 147, 147, 147, 是, 是, 是, 是, 是, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)","explanation_zh":"Wave6第九層核對=([24, 25], [3, 3], [24, 25], [3, 3], 是, 是, 147, 147, 147, 是, 是, 是, 是, 是, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性規劃對偶"]},"variant":{"family_id":"qf-mod46-lpdual-zero-gap-certificate","seed":202611154073,"parameters":{"A":[[1,0],[0,1]],"b":[24,25],"c":[3,3],"x":[24,25],"y":[3,3],"domain":"Exact_primal_dual_linear_program","operation":"lpdual-zero-gap-certificate","semantic_template":"qf-mod46-lpdual-zero-gap-certificate/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[24,1],[25,1]],\"c\":[[3,1],[3,1]],\"x\":[[24,1],[25,1]],\"y\":[[3,1],[3,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r04-linear-program-duality-0074","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-program-duality"],"title_zh":"線性規劃對偶","stem_zh":"固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-zero-gap-certificate；資料={\"A\":[[1,0],[0,1]],\"b\":[25,26],\"c\":[4,4],\"x\":[25,26],\"y\":[4,4]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([25, 26], [4, 4], [25, 26], [4, 4], 是, 是, 204, 204, 204, 是, 是, 是, 是, 是, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)","explanation_zh":"Wave6第九層核對=([25, 26], [4, 4], [25, 26], [4, 4], 是, 是, 204, 204, 204, 是, 是, 是, 是, 是, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性規劃對偶"]},"variant":{"family_id":"qf-mod46-lpdual-zero-gap-certificate","seed":202611154074,"parameters":{"A":[[1,0],[0,1]],"b":[25,26],"c":[4,4],"x":[25,26],"y":[4,4],"domain":"Exact_primal_dual_linear_program","operation":"lpdual-zero-gap-certificate","semantic_template":"qf-mod46-lpdual-zero-gap-certificate/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[25,1],[26,1]],\"c\":[[4,1],[4,1]],\"x\":[[25,1],[26,1]],\"y\":[[4,1],[4,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r04-linear-program-duality-0075","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-program-duality"],"title_zh":"線性規劃對偶","stem_zh":"固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-zero-gap-certificate；資料={\"A\":[[1,0],[0,1]],\"b\":[26,27],\"c\":[1,5],\"x\":[26,27],\"y\":[1,5]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([26, 27], [1, 5], [26, 27], [1, 5], 是, 是, 161, 161, 161, 是, 是, 是, 是, 是, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)","explanation_zh":"Wave6第九層核對=([26, 27], [1, 5], [26, 27], [1, 5], 是, 是, 161, 161, 161, 是, 是, 是, 是, 是, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性規劃對偶"]},"variant":{"family_id":"qf-mod46-lpdual-zero-gap-certificate","seed":202611154075,"parameters":{"A":[[1,0],[0,1]],"b":[26,27],"c":[1,5],"x":[26,27],"y":[1,5],"domain":"Exact_primal_dual_linear_program","operation":"lpdual-zero-gap-certificate","semantic_template":"qf-mod46-lpdual-zero-gap-certificate/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[26,1],[27,1]],\"c\":[[1,1],[5,1]],\"x\":[[26,1],[27,1]],\"y\":[[1,1],[5,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r04-linear-program-duality-0076","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-program-duality"],"title_zh":"線性規劃對偶","stem_zh":"固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-strong-boundary；資料={\"mode\":\"primal_infeasible_equal_objective\",\"A\":[[1]],\"b\":[2],\"c\":[1],\"x\":[3],\"y\":[[3,2]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(primal_infeasible_equal_objective, [3], [3/2], [3], [3/2], 否, 是, 3, 9/2, 3, 是, 否, 是, 否, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)","explanation_zh":"Wave6第九層核對=(primal_infeasible_equal_objective, [3], [3/2], [3], [3/2], 否, 是, 3, 9/2, 3, 是, 否, 是, 否, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性規劃對偶"]},"variant":{"family_id":"qf-mod46-lpdual-strong-boundary","seed":202611154076,"parameters":{"mode":"primal_infeasible_equal_objective","A":[[1]],"b":[2],"c":[1],"x":[3],"y":[[3,2]],"domain":"Exact_primal_dual_linear_program","operation":"lpdual-strong-boundary","semantic_template":"qf-mod46-lpdual-strong-boundary/{\"A\":[[[1,1]]],\"b\":[[2,1]],\"c\":[[1,1]],\"mode\":\"primal_infeasible_equal_objective\",\"x\":[[3,1]],\"y\":[[3,2]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r04-linear-program-duality-0077","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-program-duality"],"title_zh":"線性規劃對偶","stem_zh":"固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-strong-boundary；資料={\"mode\":\"dual_infeasible_equal_objective\",\"A\":[[1]],\"b\":[1],\"c\":[3],\"x\":[[1,3]],\"y\":[1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(dual_infeasible_equal_objective, [1/3], [1], [1/3], [1], 是, 否, 1, 1/3, 1, 否, 是, 是, 否, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)","explanation_zh":"Wave6第九層核對=(dual_infeasible_equal_objective, [1/3], [1], [1/3], [1], 是, 否, 1, 1/3, 1, 否, 是, 是, 否, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性規劃對偶"]},"variant":{"family_id":"qf-mod46-lpdual-strong-boundary","seed":202611154077,"parameters":{"mode":"dual_infeasible_equal_objective","A":[[1]],"b":[1],"c":[3],"x":[[1,3]],"y":[1],"domain":"Exact_primal_dual_linear_program","operation":"lpdual-strong-boundary","semantic_template":"qf-mod46-lpdual-strong-boundary/{\"A\":[[[1,1]]],\"b\":[[1,1]],\"c\":[[3,1]],\"mode\":\"dual_infeasible_equal_objective\",\"x\":[[1,3]],\"y\":[[1,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r04-linear-program-duality-0078","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-program-duality"],"title_zh":"線性規劃對偶","stem_zh":"固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-strong-boundary；資料={\"mode\":\"finite_grid_not_general\",\"A\":[[1]],\"b\":[4],\"c\":[1],\"x\":[1],\"y\":[2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(finite_grid_not_general, [1], [2], [1], [2], 是, 是, 1, 2, 8, 是, 是, 否, 否, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)","explanation_zh":"Wave6第九層核對=(finite_grid_not_general, [1], [2], [1], [2], 是, 是, 1, 2, 8, 是, 是, 否, 否, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性規劃對偶"]},"variant":{"family_id":"qf-mod46-lpdual-strong-boundary","seed":202611154078,"parameters":{"mode":"finite_grid_not_general","A":[[1]],"b":[4],"c":[1],"x":[1],"y":[2],"domain":"Exact_primal_dual_linear_program","operation":"lpdual-strong-boundary","semantic_template":"qf-mod46-lpdual-strong-boundary/{\"A\":[[[1,1]]],\"b\":[[4,1]],\"c\":[[1,1]],\"mode\":\"finite_grid_not_general\",\"x\":[[1,1]],\"y\":[[2,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r04-linear-program-duality-0079","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-program-duality"],"title_zh":"線性規劃對偶","stem_zh":"固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-strong-boundary；資料={\"mode\":\"primal_infeasible_equal_objective\",\"A\":[[1]],\"b\":[5],\"c\":[1],\"x\":[6],\"y\":[[6,5]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(primal_infeasible_equal_objective, [6], [6/5], [6], [6/5], 否, 是, 6, 36/5, 6, 是, 否, 是, 否, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)","explanation_zh":"Wave6第九層核對=(primal_infeasible_equal_objective, [6], [6/5], [6], [6/5], 否, 是, 6, 36/5, 6, 是, 否, 是, 否, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性規劃對偶"]},"variant":{"family_id":"qf-mod46-lpdual-strong-boundary","seed":202611154079,"parameters":{"mode":"primal_infeasible_equal_objective","A":[[1]],"b":[5],"c":[1],"x":[6],"y":[[6,5]],"domain":"Exact_primal_dual_linear_program","operation":"lpdual-strong-boundary","semantic_template":"qf-mod46-lpdual-strong-boundary/{\"A\":[[[1,1]]],\"b\":[[5,1]],\"c\":[[1,1]],\"mode\":\"primal_infeasible_equal_objective\",\"x\":[[6,1]],\"y\":[[6,5]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r04-linear-program-duality-0080","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-program-duality"],"title_zh":"線性規劃對偶","stem_zh":"固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-strong-boundary；資料={\"mode\":\"dual_infeasible_equal_objective\",\"A\":[[1]],\"b\":[1],\"c\":[6],\"x\":[[1,6]],\"y\":[1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(dual_infeasible_equal_objective, [1/6], [1], [1/6], [1], 是, 否, 1, 1/6, 1, 否, 是, 是, 否, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)","explanation_zh":"Wave6第九層核對=(dual_infeasible_equal_objective, [1/6], [1], [1/6], [1], 是, 否, 1, 1/6, 1, 否, 是, 是, 否, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性規劃對偶"]},"variant":{"family_id":"qf-mod46-lpdual-strong-boundary","seed":202611154080,"parameters":{"mode":"dual_infeasible_equal_objective","A":[[1]],"b":[1],"c":[6],"x":[[1,6]],"y":[1],"domain":"Exact_primal_dual_linear_program","operation":"lpdual-strong-boundary","semantic_template":"qf-mod46-lpdual-strong-boundary/{\"A\":[[[1,1]]],\"b\":[[1,1]],\"c\":[[6,1]],\"mode\":\"dual_infeasible_equal_objective\",\"x\":[[1,6]],\"y\":[[1,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r04-linear-program-duality-0081","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-program-duality"],"title_zh":"線性規劃對偶","stem_zh":"固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-strong-boundary；資料={\"mode\":\"finite_grid_not_general\",\"A\":[[1]],\"b\":[7],\"c\":[1],\"x\":[1],\"y\":[2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(finite_grid_not_general, [1], [2], [1], [2], 是, 是, 1, 2, 14, 是, 是, 否, 否, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)","explanation_zh":"Wave6第九層核對=(finite_grid_not_general, [1], [2], [1], [2], 是, 是, 1, 2, 14, 是, 是, 否, 否, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性規劃對偶"]},"variant":{"family_id":"qf-mod46-lpdual-strong-boundary","seed":202611154081,"parameters":{"mode":"finite_grid_not_general","A":[[1]],"b":[7],"c":[1],"x":[1],"y":[2],"domain":"Exact_primal_dual_linear_program","operation":"lpdual-strong-boundary","semantic_template":"qf-mod46-lpdual-strong-boundary/{\"A\":[[[1,1]]],\"b\":[[7,1]],\"c\":[[1,1]],\"mode\":\"finite_grid_not_general\",\"x\":[[1,1]],\"y\":[[2,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r04-linear-program-duality-0082","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-program-duality"],"title_zh":"線性規劃對偶","stem_zh":"固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-strong-boundary；資料={\"mode\":\"primal_infeasible_equal_objective\",\"A\":[[1]],\"b\":[8],\"c\":[1],\"x\":[9],\"y\":[[9,8]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(primal_infeasible_equal_objective, [9], [9/8], [9], [9/8], 否, 是, 9, 81/8, 9, 是, 否, 是, 否, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)","explanation_zh":"Wave6第九層核對=(primal_infeasible_equal_objective, [9], [9/8], [9], [9/8], 否, 是, 9, 81/8, 9, 是, 否, 是, 否, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性規劃對偶"]},"variant":{"family_id":"qf-mod46-lpdual-strong-boundary","seed":202611154082,"parameters":{"mode":"primal_infeasible_equal_objective","A":[[1]],"b":[8],"c":[1],"x":[9],"y":[[9,8]],"domain":"Exact_primal_dual_linear_program","operation":"lpdual-strong-boundary","semantic_template":"qf-mod46-lpdual-strong-boundary/{\"A\":[[[1,1]]],\"b\":[[8,1]],\"c\":[[1,1]],\"mode\":\"primal_infeasible_equal_objective\",\"x\":[[9,1]],\"y\":[[9,8]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r04-linear-program-duality-0083","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-program-duality"],"title_zh":"線性規劃對偶","stem_zh":"固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-strong-boundary；資料={\"mode\":\"dual_infeasible_equal_objective\",\"A\":[[1]],\"b\":[1],\"c\":[9],\"x\":[[1,9]],\"y\":[1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(dual_infeasible_equal_objective, [1/9], [1], [1/9], [1], 是, 否, 1, 1/9, 1, 否, 是, 是, 否, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)","explanation_zh":"Wave6第九層核對=(dual_infeasible_equal_objective, [1/9], [1], [1/9], [1], 是, 否, 1, 1/9, 1, 否, 是, 是, 否, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["線性規劃對偶"]},"variant":{"family_id":"qf-mod46-lpdual-strong-boundary","seed":202611154083,"parameters":{"mode":"dual_infeasible_equal_objective","A":[[1]],"b":[1],"c":[9],"x":[[1,9]],"y":[1],"domain":"Exact_primal_dual_linear_program","operation":"lpdual-strong-boundary","semantic_template":"qf-mod46-lpdual-strong-boundary/{\"A\":[[[1,1]]],\"b\":[[1,1]],\"c\":[[9,1]],\"mode\":\"dual_infeasible_equal_objective\",\"x\":[[1,9]],\"y\":[[1,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r04-linear-program-duality-0084","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-program-duality"],"title_zh":"線性規劃對偶","stem_zh":"固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-strong-boundary；資料={\"mode\":\"finite_grid_not_general\",\"A\":[[1]],\"b\":[10],\"c\":[1],\"x\":[1],\"y\":[2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(finite_grid_not_general, [1], [2], [1], [2], 是, 是, 1, 2, 20, 是, 是, 否, 否, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)","explanation_zh":"Wave6第九層核對=(finite_grid_not_general, [1], [2], [1], [2], 是, 是, 1, 2, 20, 是, 是, 否, 否, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性規劃對偶"]},"variant":{"family_id":"qf-mod46-lpdual-strong-boundary","seed":202611154084,"parameters":{"mode":"finite_grid_not_general","A":[[1]],"b":[10],"c":[1],"x":[1],"y":[2],"domain":"Exact_primal_dual_linear_program","operation":"lpdual-strong-boundary","semantic_template":"qf-mod46-lpdual-strong-boundary/{\"A\":[[[1,1]]],\"b\":[[10,1]],\"c\":[[1,1]],\"mode\":\"finite_grid_not_general\",\"x\":[[1,1]],\"y\":[[2,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r04-linear-program-duality-0085","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-program-duality"],"title_zh":"線性規劃對偶","stem_zh":"固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-strong-boundary；資料={\"mode\":\"primal_infeasible_equal_objective\",\"A\":[[1]],\"b\":[11],\"c\":[1],\"x\":[12],\"y\":[[12,11]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(primal_infeasible_equal_objective, [12], [12/11], [12], [12/11], 否, 是, 12, 144/11, 12, 是, 否, 是, 否, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)","explanation_zh":"Wave6第九層核對=(primal_infeasible_equal_objective, [12], [12/11], [12], [12/11], 否, 是, 12, 144/11, 12, 是, 否, 是, 否, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性規劃對偶"]},"variant":{"family_id":"qf-mod46-lpdual-strong-boundary","seed":202611154085,"parameters":{"mode":"primal_infeasible_equal_objective","A":[[1]],"b":[11],"c":[1],"x":[12],"y":[[12,11]],"domain":"Exact_primal_dual_linear_program","operation":"lpdual-strong-boundary","semantic_template":"qf-mod46-lpdual-strong-boundary/{\"A\":[[[1,1]]],\"b\":[[11,1]],\"c\":[[1,1]],\"mode\":\"primal_infeasible_equal_objective\",\"x\":[[12,1]],\"y\":[[12,11]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r04-linear-program-duality-0086","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-program-duality"],"title_zh":"線性規劃對偶","stem_zh":"固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-strong-boundary；資料={\"mode\":\"dual_infeasible_equal_objective\",\"A\":[[1]],\"b\":[1],\"c\":[12],\"x\":[[1,12]],\"y\":[1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(dual_infeasible_equal_objective, [1/12], [1], [1/12], [1], 是, 否, 1, 1/12, 1, 否, 是, 是, 否, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)","explanation_zh":"Wave6第九層核對=(dual_infeasible_equal_objective, [1/12], [1], [1/12], [1], 是, 否, 1, 1/12, 1, 否, 是, 是, 否, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性規劃對偶"]},"variant":{"family_id":"qf-mod46-lpdual-strong-boundary","seed":202611154086,"parameters":{"mode":"dual_infeasible_equal_objective","A":[[1]],"b":[1],"c":[12],"x":[[1,12]],"y":[1],"domain":"Exact_primal_dual_linear_program","operation":"lpdual-strong-boundary","semantic_template":"qf-mod46-lpdual-strong-boundary/{\"A\":[[[1,1]]],\"b\":[[1,1]],\"c\":[[12,1]],\"mode\":\"dual_infeasible_equal_objective\",\"x\":[[1,12]],\"y\":[[1,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r04-linear-program-duality-0087","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-program-duality"],"title_zh":"線性規劃對偶","stem_zh":"固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-strong-boundary；資料={\"mode\":\"finite_grid_not_general\",\"A\":[[1]],\"b\":[13],\"c\":[1],\"x\":[1],\"y\":[2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(finite_grid_not_general, [1], [2], [1], [2], 是, 是, 1, 2, 26, 是, 是, 否, 否, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)","explanation_zh":"Wave6第九層核對=(finite_grid_not_general, [1], [2], [1], [2], 是, 是, 1, 2, 26, 是, 是, 否, 否, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性規劃對偶"]},"variant":{"family_id":"qf-mod46-lpdual-strong-boundary","seed":202611154087,"parameters":{"mode":"finite_grid_not_general","A":[[1]],"b":[13],"c":[1],"x":[1],"y":[2],"domain":"Exact_primal_dual_linear_program","operation":"lpdual-strong-boundary","semantic_template":"qf-mod46-lpdual-strong-boundary/{\"A\":[[[1,1]]],\"b\":[[13,1]],\"c\":[[1,1]],\"mode\":\"finite_grid_not_general\",\"x\":[[1,1]],\"y\":[[2,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r04-linear-program-duality-0088","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-program-duality"],"title_zh":"線性規劃對偶","stem_zh":"固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-strong-boundary；資料={\"mode\":\"primal_infeasible_equal_objective\",\"A\":[[1]],\"b\":[14],\"c\":[1],\"x\":[15],\"y\":[[15,14]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(primal_infeasible_equal_objective, [15], [15/14], [15], [15/14], 否, 是, 15, 225/14, 15, 是, 否, 是, 否, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)","explanation_zh":"Wave6第九層核對=(primal_infeasible_equal_objective, [15], [15/14], [15], [15/14], 否, 是, 15, 225/14, 15, 是, 否, 是, 否, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性規劃對偶"]},"variant":{"family_id":"qf-mod46-lpdual-strong-boundary","seed":202611154088,"parameters":{"mode":"primal_infeasible_equal_objective","A":[[1]],"b":[14],"c":[1],"x":[15],"y":[[15,14]],"domain":"Exact_primal_dual_linear_program","operation":"lpdual-strong-boundary","semantic_template":"qf-mod46-lpdual-strong-boundary/{\"A\":[[[1,1]]],\"b\":[[14,1]],\"c\":[[1,1]],\"mode\":\"primal_infeasible_equal_objective\",\"x\":[[15,1]],\"y\":[[15,14]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r04-linear-program-duality-0089","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-program-duality"],"title_zh":"線性規劃對偶","stem_zh":"固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-strong-boundary；資料={\"mode\":\"dual_infeasible_equal_objective\",\"A\":[[1]],\"b\":[1],\"c\":[15],\"x\":[[1,15]],\"y\":[1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(dual_infeasible_equal_objective, [1/15], [1], [1/15], [1], 是, 否, 1, 1/15, 1, 否, 是, 是, 否, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)","explanation_zh":"Wave6第九層核對=(dual_infeasible_equal_objective, [1/15], [1], [1/15], [1], 是, 否, 1, 1/15, 1, 否, 是, 是, 否, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性規劃對偶"]},"variant":{"family_id":"qf-mod46-lpdual-strong-boundary","seed":202611154089,"parameters":{"mode":"dual_infeasible_equal_objective","A":[[1]],"b":[1],"c":[15],"x":[[1,15]],"y":[1],"domain":"Exact_primal_dual_linear_program","operation":"lpdual-strong-boundary","semantic_template":"qf-mod46-lpdual-strong-boundary/{\"A\":[[[1,1]]],\"b\":[[1,1]],\"c\":[[15,1]],\"mode\":\"dual_infeasible_equal_objective\",\"x\":[[1,15]],\"y\":[[1,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r04-linear-program-duality-0090","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-program-duality"],"title_zh":"線性規劃對偶","stem_zh":"固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-strong-boundary；資料={\"mode\":\"finite_grid_not_general\",\"A\":[[1]],\"b\":[16],\"c\":[1],\"x\":[1],\"y\":[2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(finite_grid_not_general, [1], [2], [1], [2], 是, 是, 1, 2, 32, 是, 是, 否, 否, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)","explanation_zh":"Wave6第九層核對=(finite_grid_not_general, [1], [2], [1], [2], 是, 是, 1, 2, 32, 是, 是, 否, 否, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性規劃對偶"]},"variant":{"family_id":"qf-mod46-lpdual-strong-boundary","seed":202611154090,"parameters":{"mode":"finite_grid_not_general","A":[[1]],"b":[16],"c":[1],"x":[1],"y":[2],"domain":"Exact_primal_dual_linear_program","operation":"lpdual-strong-boundary","semantic_template":"qf-mod46-lpdual-strong-boundary/{\"A\":[[[1,1]]],\"b\":[[16,1]],\"c\":[[1,1]],\"mode\":\"finite_grid_not_general\",\"x\":[[1,1]],\"y\":[[2,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r04-linear-program-duality-0091","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-program-duality"],"title_zh":"線性規劃對偶","stem_zh":"固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-strong-boundary；資料={\"mode\":\"primal_infeasible_equal_objective\",\"A\":[[1]],\"b\":[17],\"c\":[1],\"x\":[18],\"y\":[[18,17]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(primal_infeasible_equal_objective, [18], [18/17], [18], [18/17], 否, 是, 18, 324/17, 18, 是, 否, 是, 否, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)","explanation_zh":"Wave6第九層核對=(primal_infeasible_equal_objective, [18], [18/17], [18], [18/17], 否, 是, 18, 324/17, 18, 是, 否, 是, 否, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性規劃對偶"]},"variant":{"family_id":"qf-mod46-lpdual-strong-boundary","seed":202611154091,"parameters":{"mode":"primal_infeasible_equal_objective","A":[[1]],"b":[17],"c":[1],"x":[18],"y":[[18,17]],"domain":"Exact_primal_dual_linear_program","operation":"lpdual-strong-boundary","semantic_template":"qf-mod46-lpdual-strong-boundary/{\"A\":[[[1,1]]],\"b\":[[17,1]],\"c\":[[1,1]],\"mode\":\"primal_infeasible_equal_objective\",\"x\":[[18,1]],\"y\":[[18,17]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r04-linear-program-duality-0092","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-program-duality"],"title_zh":"線性規劃對偶","stem_zh":"固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-strong-boundary；資料={\"mode\":\"dual_infeasible_equal_objective\",\"A\":[[1]],\"b\":[1],\"c\":[18],\"x\":[[1,18]],\"y\":[1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(dual_infeasible_equal_objective, [1/18], [1], [1/18], [1], 是, 否, 1, 1/18, 1, 否, 是, 是, 否, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)","explanation_zh":"Wave6第九層核對=(dual_infeasible_equal_objective, [1/18], [1], [1/18], [1], 是, 否, 1, 1/18, 1, 否, 是, 是, 否, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["線性規劃對偶"]},"variant":{"family_id":"qf-mod46-lpdual-strong-boundary","seed":202611154092,"parameters":{"mode":"dual_infeasible_equal_objective","A":[[1]],"b":[1],"c":[18],"x":[[1,18]],"y":[1],"domain":"Exact_primal_dual_linear_program","operation":"lpdual-strong-boundary","semantic_template":"qf-mod46-lpdual-strong-boundary/{\"A\":[[[1,1]]],\"b\":[[1,1]],\"c\":[[18,1]],\"mode\":\"dual_infeasible_equal_objective\",\"x\":[[1,18]],\"y\":[[1,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r04-linear-program-duality-0093","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-program-duality"],"title_zh":"線性規劃對偶","stem_zh":"固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-strong-boundary；資料={\"mode\":\"finite_grid_not_general\",\"A\":[[1]],\"b\":[19],\"c\":[1],\"x\":[1],\"y\":[2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(finite_grid_not_general, [1], [2], [1], [2], 是, 是, 1, 2, 38, 是, 是, 否, 否, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)","explanation_zh":"Wave6第九層核對=(finite_grid_not_general, [1], [2], [1], [2], 是, 是, 1, 2, 38, 是, 是, 否, 否, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性規劃對偶"]},"variant":{"family_id":"qf-mod46-lpdual-strong-boundary","seed":202611154093,"parameters":{"mode":"finite_grid_not_general","A":[[1]],"b":[19],"c":[1],"x":[1],"y":[2],"domain":"Exact_primal_dual_linear_program","operation":"lpdual-strong-boundary","semantic_template":"qf-mod46-lpdual-strong-boundary/{\"A\":[[[1,1]]],\"b\":[[19,1]],\"c\":[[1,1]],\"mode\":\"finite_grid_not_general\",\"x\":[[1,1]],\"y\":[[2,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r04-linear-program-duality-0094","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-program-duality"],"title_zh":"線性規劃對偶","stem_zh":"固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-strong-boundary；資料={\"mode\":\"primal_infeasible_equal_objective\",\"A\":[[1]],\"b\":[20],\"c\":[1],\"x\":[21],\"y\":[[21,20]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(primal_infeasible_equal_objective, [21], [21/20], [21], [21/20], 否, 是, 21, 441/20, 21, 是, 否, 是, 否, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)","explanation_zh":"Wave6第九層核對=(primal_infeasible_equal_objective, [21], [21/20], [21], [21/20], 否, 是, 21, 441/20, 21, 是, 否, 是, 否, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性規劃對偶"]},"variant":{"family_id":"qf-mod46-lpdual-strong-boundary","seed":202611154094,"parameters":{"mode":"primal_infeasible_equal_objective","A":[[1]],"b":[20],"c":[1],"x":[21],"y":[[21,20]],"domain":"Exact_primal_dual_linear_program","operation":"lpdual-strong-boundary","semantic_template":"qf-mod46-lpdual-strong-boundary/{\"A\":[[[1,1]]],\"b\":[[20,1]],\"c\":[[1,1]],\"mode\":\"primal_infeasible_equal_objective\",\"x\":[[21,1]],\"y\":[[21,20]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r04-linear-program-duality-0095","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-program-duality"],"title_zh":"線性規劃對偶","stem_zh":"固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-strong-boundary；資料={\"mode\":\"dual_infeasible_equal_objective\",\"A\":[[1]],\"b\":[1],\"c\":[21],\"x\":[[1,21]],\"y\":[1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(dual_infeasible_equal_objective, [1/21], [1], [1/21], [1], 是, 否, 1, 1/21, 1, 否, 是, 是, 否, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)","explanation_zh":"Wave6第九層核對=(dual_infeasible_equal_objective, [1/21], [1], [1/21], [1], 是, 否, 1, 1/21, 1, 否, 是, 是, 否, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性規劃對偶"]},"variant":{"family_id":"qf-mod46-lpdual-strong-boundary","seed":202611154095,"parameters":{"mode":"dual_infeasible_equal_objective","A":[[1]],"b":[1],"c":[21],"x":[[1,21]],"y":[1],"domain":"Exact_primal_dual_linear_program","operation":"lpdual-strong-boundary","semantic_template":"qf-mod46-lpdual-strong-boundary/{\"A\":[[[1,1]]],\"b\":[[1,1]],\"c\":[[21,1]],\"mode\":\"dual_infeasible_equal_objective\",\"x\":[[1,21]],\"y\":[[1,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r04-linear-program-duality-0096","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-program-duality"],"title_zh":"線性規劃對偶","stem_zh":"固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-strong-boundary；資料={\"mode\":\"finite_grid_not_general\",\"A\":[[1]],\"b\":[22],\"c\":[1],\"x\":[1],\"y\":[2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(finite_grid_not_general, [1], [2], [1], [2], 是, 是, 1, 2, 44, 是, 是, 否, 否, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)","explanation_zh":"Wave6第九層核對=(finite_grid_not_general, [1], [2], [1], [2], 是, 是, 1, 2, 44, 是, 是, 否, 否, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性規劃對偶"]},"variant":{"family_id":"qf-mod46-lpdual-strong-boundary","seed":202611154096,"parameters":{"mode":"finite_grid_not_general","A":[[1]],"b":[22],"c":[1],"x":[1],"y":[2],"domain":"Exact_primal_dual_linear_program","operation":"lpdual-strong-boundary","semantic_template":"qf-mod46-lpdual-strong-boundary/{\"A\":[[[1,1]]],\"b\":[[22,1]],\"c\":[[1,1]],\"mode\":\"finite_grid_not_general\",\"x\":[[1,1]],\"y\":[[2,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r04-linear-program-duality-0097","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-program-duality"],"title_zh":"線性規劃對偶","stem_zh":"固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-strong-boundary；資料={\"mode\":\"primal_infeasible_equal_objective\",\"A\":[[1]],\"b\":[23],\"c\":[1],\"x\":[24],\"y\":[[24,23]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(primal_infeasible_equal_objective, [24], [24/23], [24], [24/23], 否, 是, 24, 576/23, 24, 是, 否, 是, 否, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)","explanation_zh":"Wave6第九層核對=(primal_infeasible_equal_objective, [24], [24/23], [24], [24/23], 否, 是, 24, 576/23, 24, 是, 否, 是, 否, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性規劃對偶"]},"variant":{"family_id":"qf-mod46-lpdual-strong-boundary","seed":202611154097,"parameters":{"mode":"primal_infeasible_equal_objective","A":[[1]],"b":[23],"c":[1],"x":[24],"y":[[24,23]],"domain":"Exact_primal_dual_linear_program","operation":"lpdual-strong-boundary","semantic_template":"qf-mod46-lpdual-strong-boundary/{\"A\":[[[1,1]]],\"b\":[[23,1]],\"c\":[[1,1]],\"mode\":\"primal_infeasible_equal_objective\",\"x\":[[24,1]],\"y\":[[24,23]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r04-linear-program-duality-0098","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-program-duality"],"title_zh":"線性規劃對偶","stem_zh":"固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-strong-boundary；資料={\"mode\":\"dual_infeasible_equal_objective\",\"A\":[[1]],\"b\":[1],\"c\":[24],\"x\":[[1,24]],\"y\":[1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(dual_infeasible_equal_objective, [1/24], [1], [1/24], [1], 是, 否, 1, 1/24, 1, 否, 是, 是, 否, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)","explanation_zh":"Wave6第九層核對=(dual_infeasible_equal_objective, [1/24], [1], [1/24], [1], 是, 否, 1, 1/24, 1, 否, 是, 是, 否, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性規劃對偶"]},"variant":{"family_id":"qf-mod46-lpdual-strong-boundary","seed":202611154098,"parameters":{"mode":"dual_infeasible_equal_objective","A":[[1]],"b":[1],"c":[24],"x":[[1,24]],"y":[1],"domain":"Exact_primal_dual_linear_program","operation":"lpdual-strong-boundary","semantic_template":"qf-mod46-lpdual-strong-boundary/{\"A\":[[[1,1]]],\"b\":[[1,1]],\"c\":[[24,1]],\"mode\":\"dual_infeasible_equal_objective\",\"x\":[[1,24]],\"y\":[[1,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r04-linear-program-duality-0099","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-program-duality"],"title_zh":"線性規劃對偶","stem_zh":"固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-strong-boundary；資料={\"mode\":\"finite_grid_not_general\",\"A\":[[1]],\"b\":[25],\"c\":[1],\"x\":[1],\"y\":[2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(finite_grid_not_general, [1], [2], [1], [2], 是, 是, 1, 2, 50, 是, 是, 否, 否, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)","explanation_zh":"Wave6第九層核對=(finite_grid_not_general, [1], [2], [1], [2], 是, 是, 1, 2, 50, 是, 是, 否, 否, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性規劃對偶"]},"variant":{"family_id":"qf-mod46-lpdual-strong-boundary","seed":202611154099,"parameters":{"mode":"finite_grid_not_general","A":[[1]],"b":[25],"c":[1],"x":[1],"y":[2],"domain":"Exact_primal_dual_linear_program","operation":"lpdual-strong-boundary","semantic_template":"qf-mod46-lpdual-strong-boundary/{\"A\":[[[1,1]]],\"b\":[[25,1]],\"c\":[[1,1]],\"mode\":\"finite_grid_not_general\",\"x\":[[1,1]],\"y\":[[2,1]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261115-r04-linear-program-duality-0100","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","language":"zh-Hant","source_mko_ids":["mko-linear-program-duality"],"title_zh":"線性規劃對偶","stem_zh":"固定max/≤/nonnegative primal與min/≥/nonnegative dual convention。操作=lpdual-strong-boundary；資料={\"mode\":\"primal_infeasible_equal_objective\",\"A\":[[1]],\"b\":[26],\"c\":[1],\"x\":[27],\"y\":[[27,26]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(primal_infeasible_equal_objective, [27], [27/26], [27], [27/26], 否, 是, 27, 729/26, 27, 是, 否, 是, 否, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)","explanation_zh":"Wave6第九層核對=(primal_infeasible_equal_objective, [27], [27/26], [27], [27/26], 否, 是, 27, 729/26, 27, 是, 否, 是, 否, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)；依完整明示資料獨立重算；feasible pair須逐項驗cTx≤yTAx≤bTy；雙方feasible且zero gap才是雙optimality certificate；缺feasibility的objective equality無效；finite grid不證一般strong duality或處理infeasible/unbounded cases。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["線性規劃對偶"]},"variant":{"family_id":"qf-mod46-lpdual-strong-boundary","seed":202611154100,"parameters":{"mode":"primal_infeasible_equal_objective","A":[[1]],"b":[26],"c":[1],"x":[27],"y":[[27,26]],"domain":"Exact_primal_dual_linear_program","operation":"lpdual-strong-boundary","semantic_template":"qf-mod46-lpdual-strong-boundary/{\"A\":[[[1,1]]],\"b\":[[26,1]],\"c\":[[1,1]],\"mode\":\"primal_infeasible_equal_objective\",\"x\":[[27,1]],\"y\":[[27,26]]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}