{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261015-advance-20261114-r06-linear-program-standard-form-0041",
  "pilot_number": "R0049-25066",
  "number": 25066,
  "family_id": "qf-mod45-lp-standard-slack-conversion",
  "title": {
    "zh-Hant": "第 25066 題：線性規劃標準形",
    "en": "Question 25066: The Slack Equality Representation Preserves Feasibility in Both Directions"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261015-advance-20261114-r06-linear-program-standard-form-0041",
    "batch_id": "batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-linear-program-standard-form"
    ],
    "title_zh": "線性規劃標準形",
    "stem_zh": "固定maximize cTx、Ax≤b、x≥0的唯一standard-form convention。操作=lp-standard-slack-conversion；資料={\"A\":[[16,1],[1,16]],\"b\":[19,19],\"c\":[17,1],\"x\":[1,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "([1, 1], [17, 17], [19, 19], [2, 2], 是, 是, 是, 是, 是, 18, x_maps_to_(x,b-Ax)_and_every_nonnegative_slack_equality_point_maps_back_bijectively)",
      "explanation_zh": "Wave6第八層核對=([1, 1], [17, 17], [19, 19], [2, 2], 是, 是, 是, 是, 是, 18, x_maps_to_(x,b-Ax)_and_every_nonnegative_slack_equality_point_maps_back_bijectively)；依完整明示資料獨立重算；dimensions/sign先資格核對；slack與unrestricted split給雙向point representation；≥、equality與minimize轉換不證feasible、bounded、optimal或dual，且不得混用另一convention。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 2,
      "skills": [
        "線性規劃標準形"
      ]
    },
    "variant": {
      "family_id": "qf-mod45-lp-standard-slack-conversion",
      "seed": 202611146041,
      "parameters": {
        "A": [
          [
            16,
            1
          ],
          [
            1,
            16
          ]
        ],
        "b": [
          19,
          19
        ],
        "c": [
          17,
          1
        ],
        "x": [
          1,
          1
        ],
        "domain": "Exact_max_le_nonnegative_LP_standard_form",
        "operation": "lp-standard-slack-conversion",
        "semantic_template": "qf-mod45-lp-standard-slack-conversion/{\"A\":[[[16,1],[1,1]],[[1,1],[16,1]]],\"b\":[[19,1],[19,1]],\"c\":[[17,1],[1,1]],\"x\":[[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、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、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-20261114-r06-linear-program-standard-form-0041",
    "batch_id": "batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "ce4bd0ab964882838900ade771fa742494160ef7350ec7ea49c2845297c3bb68",
      "family_id": "qf-mod45-lp-standard-slack-conversion",
      "seed": 202611146041,
      "semantic_template": "qf-mod45-lp-standard-slack-conversion/{\"A\":[[[16,1],[1,1]],[[1,1],[16,1]]],\"b\":[[19,1],[19,1]],\"c\":[[17,1],[1,1]],\"x\":[[1,1],[1,1]]}"
    },
    "fields": {
      "title": "The Slack Equality Representation Preserves Feasibility in Both Directions",
      "stem": "Use the one fixed convention maximize c^T*x, subject to A*x≤b and x≥0, with A=[[16, 1], [1, 16]], b=[19, 19], c=[17, 1]. Require a nonempty rectangular exact rational matrix and matching vectors. Coefficients and b may have either sign; variable nonnegativity is a constraint, not an assumption that every supplied point is feasible. Supplied candidate x=[1, 1]. Give (x, Ax, b, slack, x_nonnegative, inequalities_hold, slack_nonnegative, feasible, feasibility_equivalence_holds, objective_value, certificate).",
      "answer": {
        "kind": "expression",
        "value": "([1, 1], [17, 17], [19, 19], [2, 2], Yes, Yes, Yes, Yes, Yes, 18, x_maps_to_(x,b-Ax)_and_every_nonnegative_slack_equality_point_maps_back_bijectively)",
        "explanation": "Compute A*x and the unique slack s=b−A*x. Compare x≥0, A*x≤b and s≥0 separately, then joint feasibility and its equivalence with nonnegative x/slack. The equivalence flag can be Yes for an infeasible point. Projection and x→(x,b−Ax) give the slack-point bijection, preserving c^T*x; this is not an optimality proof. The exact result, retaining the requested order, is ([1, 1], [17, 17], [19, 19], [2, 2], Yes, Yes, Yes, Yes, Yes, 18, x_maps_to_(x,b-Ax)_and_every_nonnegative_slack_equality_point_maps_back_bijectively)."
      },
      "skills": [
        "The Slack Equality Representation Preserves Feasibility in Both Directions"
      ],
      "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maximize cTx、Ax≤b、x≥0的唯一standard-form convention。操作=lp-standard-slack-conversion；資料={\"A\":[[16,1],[1,16]],\"b\":[19,19],\"c\":[17,1],\"x\":[1,1]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
        "en": "Use the one fixed convention maximize c^T*x, subject to A*x≤b and x≥0, with A=[[16, 1], [1, 16]], b=[19, 19], c=[17, 1]. Require a nonempty rectangular exact rational matrix and matching vectors. Coefficients and b may have either sign; variable nonnegativity is a constraint, not an assumption that every supplied point is feasible. Supplied candidate x=[1, 1]. Give (x, Ax, b, slack, x_nonnegative, inequalities_hold, slack_nonnegative, feasible, feasibility_equivalence_holds, objective_value, certificate)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source roles = [lp-standard-slack-conversion]",
      "after": "derived = [[x, Ax, b, slack, x_nonnegative, inequalities_hold, slack_nonnegative, feasible, feasibility_equivalence_holds, objective_value, certificate]]",
      "relation": "implies",
      "rule_id": "source_ordered_fields_without_JS",
      "conditions": [],
      "explanation": {
        "zh-Hant": "本頁列完整來源欄序，不需JavaScript；LP三種mode雖同為6欄，角色仍由原mode選，不由答案形狀猜。",
        "en": "List the full source field order without JavaScript.LP boundary modes all have6 fields but select different roles from the original mode,not answer shape."
      },
      "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 = [[[16, 1], [1, 16]], [19, 19], [17, 1]]",
      "after": "derived = [2, 2, maximize cTx subject Ax<=b,x>=0, 是]",
      "relation": "implies",
      "rule_id": "LP_fixed_max_le_nonnegative_signature_NOT_feasibility",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐行核矩形與b/c維度；這是指定的standard-form慣例而非所有教科書唯一用法。A、b、c可帶任意合法符號；dim合格不證可行。",
        "en": "Check rectangular rows and b/c dimensions.This is one declared standard-form convention,not the only textbook convention.A,b,c may have either sign;qualified dimensions do not prove 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": "LP ORIGINAL constraint row0 product0 := mul(16, 1)",
      "after": "LP ORIGINAL constraint row0 product0 = 16",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值，不以小數近似或單次觀察替代。",
        "en": "Substitute original roles and established intermediates exactly,without rounding or substituting a single observation."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 LP ORIGINAL constraint row0 product0，不以小數近似替代。",
          "en": "Recompute LP ORIGINAL constraint row0 product0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "16",
          "1"
        ],
        "output": "16",
        "label": "LP ORIGINAL constraint row0 product0"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "LP ORIGINAL constraint row0 cumulative0 := sum(0, 16)",
      "after": "LP ORIGINAL constraint row0 cumulative0 = 16",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值，不以小數近似或單次觀察替代。",
        "en": "Substitute original roles and established intermediates exactly,without rounding or substituting a single observation."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 LP ORIGINAL constraint row0 cumulative0，不以小數近似替代。",
          "en": "Recompute LP ORIGINAL constraint row0 cumulative0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "16"
        ],
        "output": "16",
        "label": "LP ORIGINAL constraint row0 cumulative0"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "LP ORIGINAL constraint row0 product1 := mul(1, 1)",
      "after": "LP ORIGINAL constraint row0 product1 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值，不以小數近似或單次觀察替代。",
        "en": "Substitute original roles and established intermediates exactly,without rounding or substituting a single observation."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 LP ORIGINAL constraint row0 product1，不以小數近似替代。",
          "en": "Recompute LP ORIGINAL constraint row0 product1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "1"
        ],
        "output": "1",
        "label": "LP ORIGINAL constraint row0 product1"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "LP ORIGINAL constraint row0 cumulative1 := sum(16, 1)",
      "after": "LP ORIGINAL constraint row0 cumulative1 = 17",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值，不以小數近似或單次觀察替代。",
        "en": "Substitute original roles and established intermediates exactly,without rounding or substituting a single observation."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 LP ORIGINAL constraint row0 cumulative1，不以小數近似替代。",
          "en": "Recompute LP ORIGINAL constraint row0 cumulative1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "16",
          "1"
        ],
        "output": "17",
        "label": "LP ORIGINAL constraint row0 cumulative1"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "LP ORIGINAL constraint row1 product0 := mul(1, 1)",
      "after": "LP ORIGINAL constraint row1 product0 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值，不以小數近似或單次觀察替代。",
        "en": "Substitute original roles and established intermediates exactly,without rounding or substituting a single observation."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 LP ORIGINAL constraint row1 product0，不以小數近似替代。",
          "en": "Recompute LP ORIGINAL constraint row1 product0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "1"
        ],
        "output": "1",
        "label": "LP ORIGINAL constraint row1 product0"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "LP ORIGINAL constraint row1 cumulative0 := sum(0, 1)",
      "after": "LP ORIGINAL constraint row1 cumulative0 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值，不以小數近似或單次觀察替代。",
        "en": "Substitute original roles and established intermediates exactly,without rounding or substituting a single observation."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 LP ORIGINAL constraint row1 cumulative0，不以小數近似替代。",
          "en": "Recompute LP ORIGINAL constraint row1 cumulative0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "1"
        ],
        "output": "1",
        "label": "LP ORIGINAL constraint row1 cumulative0"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "LP ORIGINAL constraint row1 product1 := mul(16, 1)",
      "after": "LP ORIGINAL constraint row1 product1 = 16",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值，不以小數近似或單次觀察替代。",
        "en": "Substitute original roles and established intermediates exactly,without rounding or substituting a single observation."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 LP ORIGINAL constraint row1 product1，不以小數近似替代。",
          "en": "Recompute LP ORIGINAL constraint row1 product1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "16",
          "1"
        ],
        "output": "16",
        "label": "LP ORIGINAL constraint row1 product1"
      }
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "LP ORIGINAL constraint row1 cumulative1 := sum(1, 16)",
      "after": "LP ORIGINAL constraint row1 cumulative1 = 17",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值，不以小數近似或單次觀察替代。",
        "en": "Substitute original roles and established intermediates exactly,without rounding or substituting a single observation."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 LP ORIGINAL constraint row1 cumulative1，不以小數近似替代。",
          "en": "Recompute LP ORIGINAL constraint row1 cumulative1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1",
          "16"
        ],
        "output": "17",
        "label": "LP ORIGINAL constraint row1 cumulative1"
      }
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "LP UNIQUE slack row0 := sub(19, 17)",
      "after": "LP UNIQUE slack row0 = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值，不以小數近似或單次觀察替代。",
        "en": "Substitute original roles and established intermediates exactly,without rounding or substituting a single observation."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 LP UNIQUE slack row0，不以小數近似替代。",
          "en": "Recompute LP UNIQUE slack row0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "19",
          "17"
        ],
        "output": "2",
        "label": "LP UNIQUE slack row0"
      }
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "LP UNIQUE slack row1 := sub(19, 17)",
      "after": "LP UNIQUE slack row1 = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值，不以小數近似或單次觀察替代。",
        "en": "Substitute original roles and established intermediates exactly,without rounding or substituting a single observation."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 LP UNIQUE slack row1，不以小數近似替代。",
          "en": "Recompute LP UNIQUE slack row1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "19",
          "17"
        ],
        "output": "2",
        "label": "LP UNIQUE slack row1"
      }
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "LP ORIGINAL objective product0 := mul(17, 1)",
      "after": "LP ORIGINAL objective product0 = 17",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值，不以小數近似或單次觀察替代。",
        "en": "Substitute original roles and established intermediates exactly,without rounding or substituting a single observation."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 LP ORIGINAL objective product0，不以小數近似替代。",
          "en": "Recompute LP ORIGINAL objective product0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "17",
          "1"
        ],
        "output": "17",
        "label": "LP ORIGINAL objective product0"
      }
    },
    {
      "id": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "LP ORIGINAL objective cumulative0 := sum(0, 17)",
      "after": "LP ORIGINAL objective cumulative0 = 17",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值，不以小數近似或單次觀察替代。",
        "en": "Substitute original roles and established intermediates exactly,without rounding or substituting a single observation."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 LP ORIGINAL objective cumulative0，不以小數近似替代。",
          "en": "Recompute LP ORIGINAL objective cumulative0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "17"
        ],
        "output": "17",
        "label": "LP ORIGINAL objective cumulative0"
      }
    },
    {
      "id": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "LP ORIGINAL objective product1 := mul(1, 1)",
      "after": "LP ORIGINAL objective product1 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值，不以小數近似或單次觀察替代。",
        "en": "Substitute original roles and established intermediates exactly,without rounding or substituting a single observation."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 LP ORIGINAL objective product1，不以小數近似替代。",
          "en": "Recompute LP ORIGINAL objective product1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "1"
        ],
        "output": "1",
        "label": "LP ORIGINAL objective product1"
      }
    },
    {
      "id": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "LP ORIGINAL objective cumulative1 := sum(17, 1)",
      "after": "LP ORIGINAL objective cumulative1 = 18",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值，不以小數近似或單次觀察替代。",
        "en": "Substitute original roles and established intermediates exactly,without rounding or substituting a single observation."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 LP ORIGINAL objective cumulative1，不以小數近似替代。",
          "en": "Recompute LP ORIGINAL objective cumulative1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "17",
          "1"
        ],
        "output": "18",
        "label": "LP ORIGINAL objective cumulative1"
      }
    },
    {
      "id": "S017",
      "uses": [
        "G001",
        "S016"
      ],
      "before": "source roles = [[1, 1], [17, 17], [19, 19], [2, 2], [17, 1]]",
      "after": "derived = [是, 是, 是, 是, 是, 18]",
      "relation": "implies",
      "rule_id": "LP_ALL_point_checks_and_GENERAL_unique_slack_bijection",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每行Ax<=b iff b−Ax>=0，須另合取x>=0。x↦(x,b−Ax)與投影互逆：若Ax+s=b則s唯一等於b−Ax；加零slack目標係數保cTx。等價旗標即使點不可行仍可True，不是最優證書。",
        "en": "Each row Ax<=b is equivalent to b−Ax>=0,but x>=0 is a separate conjunct.The maps x->(x,b−Ax) and projection are inverses because Ax+s=b uniquely determines s.Zero slack-objective coefficients preserve cTx.The equivalence flag can be True for an infeasible point;it is not optimality."
      },
      "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": [
      [
        "1",
        "1"
      ],
      [
        "17",
        "17"
      ],
      [
        "19",
        "19"
      ],
      [
        "2",
        "2"
      ],
      true,
      true,
      true,
      true,
      true,
      "18",
      "x_maps_to_(x,b-Ax)_and_every_nonnegative_slack_equality_point_maps_back_bijectively"
    ]
  },
  "conclusion": {
    "canonical_value": "([1, 1], [17, 17], [19, 19], [2, 2], 是, 是, 是, 是, 是, 18, x_maps_to_(x,b-Ax)_and_every_nonnegative_slack_equality_point_maps_back_bijectively)",
    "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-20261114-r06-linear-program-standard-form-foundations",
      "batch_path": "public/data/questions/batches/batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations/questions.jsonl",
      "batch_sha256": "2c5c4e4aa189e219df60939f3d3fe9dde28bd1bbba58fac6d75b360f38625739",
      "original_line": 41,
      "original_line_sha256": "90f285c94a3d3d1e93a9bd0d6f11045dc9dee7278ca198238a9f66750d6980df",
      "question_compact_sha256": "ce4bd0ab964882838900ade771fa742494160ef7350ec7ea49c2845297c3bb68",
      "english_row_compact_sha256": "3afd922fd68a26c41e00561fdf194c052f89bfcd133d71d57206a32c874277f5",
      "english_manifest_path": "public/data/i18n/editions/batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-linear-program-standard-form": {
        "sha256": "cd8b89ab19acb18f321ca46da48146c104e7dad057ad394dffacf6e741ae1314",
        "english_sha256": "2ffa58673107872d4017597a54a07e42ca18a3138d345cea55419ed0c6a330ed",
        "original_path": "public/data/mko/linear-program-standard-form.json",
        "english_path": "public/data/i18n/editions/mko-linear-program-standard-form.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "991c73df066352bfcfb1142a2492e104cd1d20dafe7e9ecea8cff26a4f9d8e50",
    "original_candidate_sha256": "2c2b83f221cda5fb663ba50a3fc2c0cba62f45c62908d9302f1054626c346d69",
    "original_candidate_line_sha256": "2c2b83f221cda5fb663ba50a3fc2c0cba62f45c62908d9302f1054626c346d69"
  },
  "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-p50",
    "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": "6212b8b0b3b273f3a514e26ebd17d7e81219c56a8f47e40047ee2b6bf01d1ba3",
      "question_first_witness_sha256": "f32a23d5e1b66c7794c60def58741bcdfacff24f3cdfc9bc48109c8f71543cf3"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 17,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "ddffe2a1c04c130b58f69ffd44c15fd7c97bd106580230c08d450f6ba03cc1e8",
      "arithmetic_steps": 14
    },
    "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."
  }
}