{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261015-advance-20261115-r04-linear-program-duality-0021", "pilot_number": "R0050-25821", "number": 25821, "family_id": "qf-mod46-lpdual-construction-qualification", "title": { "zh-Hant": "第 25821 題：線性規劃對偶", "en": "Question 25821: Construct the Dual from One Fixed Primal Convention and the Exact Transpose" }, "original_question": { "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" } }, "original_english_edition": { "schema_version": "ocme-question-localization-v0.1", "question_id": "q-20261015-advance-20261115-r04-linear-program-duality-0021", "batch_id": "batch-20261015-advance-20261115-r04-linear-program-duality-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "a997ca4e5ebbdeb73bc7df935af4fe311886ce222ee7155fa8ad73a4883d7c0d", "family_id": "qf-mod46-lpdual-construction-qualification", "seed": 202611154021, "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]]}" }, "fields": { "title": "Construct the Dual from One Fixed Primal Convention and the Exact Transpose", "stem": "Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[22, 23], c=[1, 1]. Give (constraint_count, variable_count, A, b, c, transpose_A, primal_convention, dual_convention, dimensions_qualified, certificate).", "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, Yes, dimensions_signs_and_transpose_are_fixed_by_one_convention)", "explanation": "Check rectangular dimensions m,n and matching vectors. The dual has m variables and n lower-bound constraints with transposed A. Coefficients may have either sign; dimensions and signs do not certify feasibility or optimality. The exact result, retaining the requested order, is (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, Yes, dimensions_signs_and_transpose_are_fixed_by_one_convention)." }, "skills": [ "Construct the Dual from One Fixed Primal Convention and the Exact Transpose" ], "verification_warning": "This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review." } }, "givens": [ { "id": "G001", "text": { "zh-Hant": "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。", "en": "Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[22, 23], c=[1, 1]. Give (constraint_count, variable_count, A, b, c, transpose_A, primal_convention, dual_convention, dimensions_qualified, certificate)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source roles = [lpdual-construction-qualification]", "after": "derived = [[constraint_count, variable_count, A, b, c, transpose_A, primal_convention, dual_convention, dimensions_qualified, certificate]]", "relation": "implies", "rule_id": "source_ordered_fields_without_JS", "conditions": [], "explanation": { "zh-Hant": "直接列完整來源欄序，不需JavaScript。LP weak為13欄、zero與strong為15欄，前提與角色依原operation／mode，不補欄。", "en": "List the full source field order without JavaScript.LP weak uses13 fields,zero/strong15;premises and roles follow the original operation/mode without padding." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "source roles = [[[1, 0], [0, 1]], [22, 23], [1, 1]]", "after": "derived = [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]", "relation": "implies", "rule_id": "LPdual_fixed_conventions_complete_transpose_and_dimensions", "conditions": [], "explanation": { "zh-Hant": "同一慣例max/<=/x>=0對min/>=/y>=0；dual有m個變數與n條限制。逐位置轉置，signed coefficients合法，維度合格不保可行。", "en": "The fixed max/<=/x>=0 convention pairs with min/>=/y>=0.The dual has m variables and n constraints.Transpose every position;either coefficient sign is legal and dimensions do not imply feasibility." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null } ], "math_model": { "result": [ 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", true, "dimensions_signs_and_transpose_are_fixed_by_one_convention" ] }, "conclusion": { "canonical_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-Hant": "依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。", "en": "The full substitutions and premise checks give the original answer; source identity,format and status are unchanged." } }, "definition_note": { "zh-Hant": "沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。", "en": "Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate." }, "source": { "input_commit": "0f7be61e2c22317a4075561dc822c87d881a50db", "question_binding": { "batch_id": "batch-20261015-advance-20261115-r04-linear-program-duality-foundations", "batch_path": "public/data/questions/batches/batch-20261015-advance-20261115-r04-linear-program-duality-foundations/questions.jsonl", "batch_sha256": "6955402541755f47f9709a7c054671eeb426c9fd8380f4fa4b819ef31ababb52", "original_line": 21, "original_line_sha256": "306de0bdb5b8fa5316d7079512b90b34697cdd0e1f4ae9d6b3a17ee6386c9dc1", "question_compact_sha256": "a997ca4e5ebbdeb73bc7df935af4fe311886ce222ee7155fa8ad73a4883d7c0d", "english_row_compact_sha256": "af29fe61200c00de2670967b05b75d5a065fb215c3523b80cb75ba4d1a019998", "english_manifest_path": "public/data/i18n/editions/batch-20261015-advance-20261115-r04-linear-program-duality-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261015-advance-20261115-r04-linear-program-duality-foundations.en.jsonl" }, "mko_bindings": { "mko-linear-program-duality": { "sha256": "89eec94cb92d6b32f73da15c2cad46fdeedeb59b91505437cded6bef94a6a6d5", "english_sha256": "9c16a09fc23a4b6c83b28e225509ab5ae11db44820a9ebfb53fd29aa70646cb2", "original_path": "public/data/mko/source13-20261018-linear-program-duality.json", "english_path": "public/data/i18n/editions/mko-linear-program-duality.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "4624c29c8cf22048332daea19f430f922e032c30410aa690ba3f1924d8b9ee45", "original_candidate_sha256": "9659ea854e209ec271a1a0c02add4d0f78a3e8bc1fbf1d5d66997181355c467a", "original_candidate_line_sha256": "9659ea854e209ec271a1a0c02add4d0f78a3e8bc1fbf1d5d66997181355c467a" }, "historical_source_state": { "review_state": { "derivation": "candidate_authored_and_locally_checked", "source_question_state": "candidate", "independent_AI_or_human_review": "NOT_PERFORMED", "formalization": "not_formalized", "Lean": "NOT_RUN" }, "scope_note": { "zh-Hant": "來源綁定教學推導候選；精確核對不等於獨立AI／人工審查或一般形式證明。原題、答案、MKO、公式、AST與歷史不變。", "en": "Source-bound pedagogical derivation candidate.Exact checking is not independent AI/human review or general formal proof.Original questions,answers,MKOs,formulas,ASTs and history remain unchanged." }, "label": "Frozen producer metadata; current per-question review is separate below." }, "revision": { "id": "review-20261026-010-p51", "review_date": "2026-10-26", "owner_authorized_reader_publication": true, "scope": "One hundred sixty family prototypes and their mathematical/bilingual reasoning variants, stems, boundary statements and every distinct explanation shape were reviewed in a context independent of the producer; parameter substitutions, source bindings, full math models, both answer strings and every arithmetic step were independently reconstructed per question from the original parameters only, with negative controls. This grouped method does not claim a separate literal AI reading of every repeated sentence or an extra AI voter.", "review_receipts": { "semantic_review_sha256": "74c63a2c12285a3b2073f8d9f102bc2ba5fb7ed890c5b7c3b8afa94f032eda82", "question_first_witness_sha256": "6d56428610f8e86c45fd00a2ad3b27ad9c052956946f0b91d71a68536f79d95f" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 2, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "7738f6fb581d387ddb8142b72b9e77e1093a7f4eb161ecf9f12a35c55a51b88b", "arithmetic_steps": 0 }, "uses_answer_as_premise": false, "human_review": "NOT_PERFORMED", "formalization": "not_formalized", "Lean": "NOT_RUN", "Canonical_truth_promotion": false, "authority": "Owner-authorized reviewed reader publication; original question/candidate states remain historical and unchanged." } }