{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261015-advance-20261115-r04-linear-program-duality-0087",
  "pilot_number": "R0050-25837",
  "number": 25837,
  "family_id": "qf-mod46-lpdual-strong-boundary",
  "title": {
    "zh-Hant": "第 25837 題：線性規劃對偶",
    "en": "Question 25837: Equal Objective Values and Finite Grids Do Not Replace Theorem Premises"
  },
  "original_question": {
    "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"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261015-advance-20261115-r04-linear-program-duality-0087",
    "batch_id": "batch-20261015-advance-20261115-r04-linear-program-duality-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "a3e4524f3267231becc670bb05b8ae67da6ec7e832e61417b5c7bc18b8db9547",
      "family_id": "qf-mod46-lpdual-strong-boundary",
      "seed": 202611154087,
      "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]]}"
    },
    "fields": {
      "title": "Equal Objective Values and Finite Grids Do Not Replace Theorem Premises",
      "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]], b=[13], c=[1]. Candidate x=[1], y=[2]. Boundary mode=finite_grid_not_general. Give (mode, x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, unsupported_optimality_or_general_strong_duality_claim, boundary).",
      "answer": {
        "kind": "expression",
        "value": "(finite_grid_not_general, [1], [2], [1], [2], Yes, Yes, 1, 2, 26, Yes, Yes, No, No, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)",
        "explanation": "Retain each feasibility flag, both inequalities and objective equality separately. Infeasible-equal-objective modes reject an optimality inference; finite_grid_not_general rejects treating finite checking as a general strong-duality proof, even when a particular feasible zero-gap pair happens to be optimal. Infeasible/unbounded classifications require additional theorem-level work. The exact result, retaining the requested order, is (finite_grid_not_general, [1], [2], [1], [2], Yes, Yes, 1, 2, 26, Yes, Yes, No, No, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)."
      },
      "skills": [
        "Equal Objective Values and Finite Grids Do Not Replace Theorem Premises"
      ],
      "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-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。",
        "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]], b=[13], c=[1]. Candidate x=[1], y=[2]. Boundary mode=finite_grid_not_general. Give (mode, x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, unsupported_optimality_or_general_strong_duality_claim, boundary)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source roles = [lpdual-strong-boundary]",
      "after": "derived = [[mode, x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, unsupported_optimality_or_general_strong_duality_claim, boundary]]",
      "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]], [13], [1]]",
      "after": "derived = [1, 1, [[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
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "LPdual ORIGINAL Ax row0 product0 := mul(1, 1)",
      "after": "LPdual ORIGINAL Ax row0 product0 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用浮點近似；額外教學恆等式不另縮限原題數值域。",
        "en": "Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 LPdual ORIGINAL Ax row0 product0，不以小數近似替代。",
          "en": "Recompute LPdual ORIGINAL Ax row0 product0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "1"
        ],
        "output": "1",
        "label": "LPdual ORIGINAL Ax row0 product0"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "LPdual ORIGINAL Ax row0 sum := sum(1)",
      "after": "LPdual ORIGINAL Ax row0 sum = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用浮點近似；額外教學恆等式不另縮限原題數值域。",
        "en": "Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 LPdual ORIGINAL Ax row0 sum，不以小數近似替代。",
          "en": "Recompute LPdual ORIGINAL Ax row0 sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1"
        ],
        "output": "1",
        "label": "LPdual ORIGINAL Ax row0 sum"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "LPdual ORIGINAL ATy row0 product0 := mul(1, 2)",
      "after": "LPdual ORIGINAL ATy row0 product0 = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用浮點近似；額外教學恆等式不另縮限原題數值域。",
        "en": "Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 LPdual ORIGINAL ATy row0 product0，不以小數近似替代。",
          "en": "Recompute LPdual ORIGINAL ATy row0 product0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "2"
        ],
        "output": "2",
        "label": "LPdual ORIGINAL ATy row0 product0"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "LPdual ORIGINAL ATy row0 sum := sum(2)",
      "after": "LPdual ORIGINAL ATy row0 sum = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用浮點近似；額外教學恆等式不另縮限原題數值域。",
        "en": "Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 LPdual ORIGINAL ATy row0 sum，不以小數近似替代。",
          "en": "Recompute LPdual ORIGINAL ATy row0 sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "2"
        ],
        "output": "2",
        "label": "LPdual ORIGINAL ATy row0 sum"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "LPdual ORIGINAL primal objective product0 := mul(1, 1)",
      "after": "LPdual ORIGINAL primal objective product0 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用浮點近似；額外教學恆等式不另縮限原題數值域。",
        "en": "Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 LPdual ORIGINAL primal objective product0，不以小數近似替代。",
          "en": "Recompute LPdual ORIGINAL primal objective product0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "1"
        ],
        "output": "1",
        "label": "LPdual ORIGINAL primal objective product0"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "LPdual ORIGINAL primal objective sum := sum(1)",
      "after": "LPdual ORIGINAL primal objective sum = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用浮點近似；額外教學恆等式不另縮限原題數值域。",
        "en": "Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 LPdual ORIGINAL primal objective sum，不以小數近似替代。",
          "en": "Recompute LPdual ORIGINAL primal objective sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1"
        ],
        "output": "1",
        "label": "LPdual ORIGINAL primal objective sum"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "LPdual ORIGINAL middle yTAx product0 := mul(2, 1)",
      "after": "LPdual ORIGINAL middle yTAx product0 = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用浮點近似；額外教學恆等式不另縮限原題數值域。",
        "en": "Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 LPdual ORIGINAL middle yTAx product0，不以小數近似替代。",
          "en": "Recompute LPdual ORIGINAL middle yTAx product0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "1"
        ],
        "output": "2",
        "label": "LPdual ORIGINAL middle yTAx product0"
      }
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "LPdual ORIGINAL middle yTAx sum := sum(2)",
      "after": "LPdual ORIGINAL middle yTAx sum = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用浮點近似；額外教學恆等式不另縮限原題數值域。",
        "en": "Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 LPdual ORIGINAL middle yTAx sum，不以小數近似替代。",
          "en": "Recompute LPdual ORIGINAL middle yTAx sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "2"
        ],
        "output": "2",
        "label": "LPdual ORIGINAL middle yTAx sum"
      }
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "LPdual ORIGINAL dual objective product0 := mul(13, 2)",
      "after": "LPdual ORIGINAL dual objective product0 = 26",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用浮點近似；額外教學恆等式不另縮限原題數值域。",
        "en": "Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 LPdual ORIGINAL dual objective product0，不以小數近似替代。",
          "en": "Recompute LPdual ORIGINAL dual objective product0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "13",
          "2"
        ],
        "output": "26",
        "label": "LPdual ORIGINAL dual objective product0"
      }
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "LPdual ORIGINAL dual objective sum := sum(26)",
      "after": "LPdual ORIGINAL dual objective sum = 26",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用浮點近似；額外教學恆等式不另縮限原題數值域。",
        "en": "Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 LPdual ORIGINAL dual objective sum，不以小數近似替代。",
          "en": "Recompute LPdual ORIGINAL dual objective sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "26"
        ],
        "output": "26",
        "label": "LPdual ORIGINAL dual objective sum"
      }
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "source roles = [[1], [2], [1], [2], [13], [1]]",
      "after": "derived = [是, 是, 1, 2, 26, [[0, 0, 2, 1, 1, 2]], 2, [[0, 1, 1, 1]], 1, [[0, 2, 12, 24]], 24, 是, 是, 否]",
      "relation": "implies",
      "rule_id": "LPdual_ALL_feasibilities_exact_pairings_residuals_and_conditional_weak_chain",
      "conditions": [],
      "explanation": {
        "zh-Hant": "兩feasibility分開核。雙和展開證(ATy)Tx=yTAx；第一gap=sum x_j((ATy)_j−c_j)，第二gap=sum y_i(b_i−(Ax)_i)。雙方可行才由各非負因子推cTx<=middle<=bTy；若缺前提，數字不等式可能巧合，不當一般證書。教學residual不另套source未執行的新數值界。",
        "en": "Check both feasibilities separately.Expanding the double sum proves(ATy)Tx=yTAx.The first gap is sum x_j((ATy)_j−c_j),the second sum y_i(b_i−(Ax)_i).Only with both feasibility premises do nonnegative factors establish the weak chain.With missing premises,numerical inequalities may be accidental.Teaching residuals impose no new source numeric bounds."
      },
      "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": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "source roles = [finite_grid_not_general, 是, 是, 1, 26, 否]",
      "after": "derived = [否]",
      "relation": "implies",
      "rule_id": "LPdual_original_boundary_mode_premises_NOT_bare_equality_general_proof",
      "conditions": [],
      "explanation": {
        "zh-Hant": "primal-infeasible模式不額外要求dual可行；dual-infeasible模式須primal可行；finite-grid模式須雙方可行但gap可非0。固定No拒欠前提的optimality推論或一般strong-duality證明，不否定特定feasible zero-gap pair真最優。",
        "en": "Primal-infeasible mode does not additionally require dual feasibility;dual-infeasible mode requires primal feasibility;finite-grid mode requires both but may have nonzero gap.No rejects unsupported optimality/general strong-duality inference,not actual optimality of a particular feasible zero-gap pair."
      },
      "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": [
      "finite_grid_not_general",
      [
        "1"
      ],
      [
        "2"
      ],
      [
        "1"
      ],
      [
        "2"
      ],
      true,
      true,
      "1",
      "2",
      "26",
      true,
      true,
      false,
      false,
      "objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases"
    ]
  },
  "conclusion": {
    "canonical_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-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": 87,
      "original_line_sha256": "5fc3c5c8b4e9c4326ba387d95eeb18b4efe5535486dfda6463ecbe6d24e656e3",
      "question_compact_sha256": "a3e4524f3267231becc670bb05b8ae67da6ec7e832e61417b5c7bc18b8db9547",
      "english_row_compact_sha256": "c84489a2ae91b1e43dbf34e30e3e67d2a0cee4b2c71249ad3ec031d9f6eb8623",
      "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": "2235935b14e522caf08d5a573f77ddfd716e3dc45313d37fea99f931bc366773",
    "original_candidate_line_sha256": "2235935b14e522caf08d5a573f77ddfd716e3dc45313d37fea99f931bc366773"
  },
  "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": 14,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "2f8c5a88720d5bf6e6b488a3104f7ecf26878eb5a179387ef96f1e2f26e10720",
      "arithmetic_steps": 10
    },
    "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."
  }
}