{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261015-advance-20261114-r06-linear-program-standard-form-0068",
  "pilot_number": "R0049-25093",
  "number": 25093,
  "family_id": "qf-mod45-lp-standard-unrestricted-split",
  "title": {
    "zh-Hant": "第 25093 題：線性規劃標準形",
    "en": "Question 25093: A Free Variable Has a Nonnegative Difference Representation, Not a Unique Split"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261015-advance-20261114-r06-linear-program-standard-form-0068",
    "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-unrestricted-split；資料={\"column\":[18,-19,20],\"objective_coefficient\":21,\"value\":-18}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "([18, -19, 20], 21, -18, 0, 18, -18, [-324, 342, -360], [-324, 342, -360], 是, -378, -378, 是, x=x_plus-x_minus_with_both_nonnegative_preserves_column_and_objective_contribution)",
      "explanation_zh": "Wave6第八層核對=([18, -19, 20], 21, -18, 0, 18, -18, [-324, 342, -360], [-324, 342, -360], 是, -378, -378, 是, x=x_plus-x_minus_with_both_nonnegative_preserves_column_and_objective_contribution)；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": 3,
      "skills": [
        "線性規劃標準形"
      ]
    },
    "variant": {
      "family_id": "qf-mod45-lp-standard-unrestricted-split",
      "seed": 202611146068,
      "parameters": {
        "column": [
          18,
          -19,
          20
        ],
        "objective_coefficient": 21,
        "value": -18,
        "domain": "Exact_max_le_nonnegative_LP_standard_form",
        "operation": "lp-standard-unrestricted-split",
        "semantic_template": "qf-mod45-lp-standard-unrestricted-split/{\"column\":[[18,1],[-19,1],[20,1]],\"objective_coefficient\":[21,1],\"value\":[-18,1]}"
      }
    },
    "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-0068",
    "batch_id": "batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "ed9e46ac5ca0e4eb8aa13ca9e5b85f5bdc7cc184ce9c167b04ac290aa129426e",
      "family_id": "qf-mod45-lp-standard-unrestricted-split",
      "seed": 202611146068,
      "semantic_template": "qf-mod45-lp-standard-unrestricted-split/{\"column\":[[18,1],[-19,1],[20,1]],\"objective_coefficient\":[21,1],\"value\":[-18,1]}"
    },
    "fields": {
      "title": "A Free Variable Has a Nonnegative Difference Representation, Not a Unique Split",
      "stem": "Use column=[18, -19, 20], objective coefficient=21, unrestricted value=-18. Give (column, objective_coefficient, value, x_plus, x_minus, reconstructed_value, original_column_contribution, split_column_contribution, columns_equal, original_objective_contribution, split_objective_contribution, objectives_equal, certificate).",
      "answer": {
        "kind": "expression",
        "value": "([18, -19, 20], 21, -18, 0, 18, -18, [-324, 342, -360], [-324, 342, -360], Yes, -378, -378, Yes, x=x_plus-x_minus_with_both_nonnegative_preserves_column_and_objective_contribution)",
        "explanation": "Choose canonical x_plus=max(value,0), x_minus=max(−value,0), reconstruct value=x_plus−x_minus and compare original/split column and objective contributions. Reverse projection is valid, but adding the same nonnegative amount to both parts gives other representations, so the unrestricted split is not a bijection of all pairs. The exact result, retaining the requested order, is ([18, -19, 20], 21, -18, 0, 18, -18, [-324, 342, -360], [-324, 342, -360], Yes, -378, -378, Yes, x=x_plus-x_minus_with_both_nonnegative_preserves_column_and_objective_contribution)."
      },
      "skills": [
        "A Free Variable Has a Nonnegative Difference Representation, Not a Unique Split"
      ],
      "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-unrestricted-split；資料={\"column\":[18,-19,20],\"objective_coefficient\":21,\"value\":-18}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
        "en": "Use column=[18, -19, 20], objective coefficient=21, unrestricted value=-18. Give (column, objective_coefficient, value, x_plus, x_minus, reconstructed_value, original_column_contribution, split_column_contribution, columns_equal, original_objective_contribution, split_objective_contribution, objectives_equal, certificate)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source roles = [lp-standard-unrestricted-split]",
      "after": "derived = [[column, objective_coefficient, value, x_plus, x_minus, reconstructed_value, original_column_contribution, split_column_contribution, columns_equal, original_objective_contribution, split_objective_contribution, objectives_equal, 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": "LP unrestricted reconstruct value := sub(0, 18)",
      "after": "LP unrestricted reconstruct value = -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 unrestricted reconstruct value，不以小數近似替代。",
          "en": "Recompute LP unrestricted reconstruct value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "0",
          "18"
        ],
        "output": "-18",
        "label": "LP unrestricted reconstruct value"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "LP unrestricted ORIGINAL column0 := mul(18, -18)",
      "after": "LP unrestricted ORIGINAL column0 = -324",
      "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 unrestricted ORIGINAL column0，不以小數近似替代。",
          "en": "Recompute LP unrestricted ORIGINAL column0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "18",
          "-18"
        ],
        "output": "-324",
        "label": "LP unrestricted ORIGINAL column0"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "LP unrestricted positive column0 := mul(18, 0)",
      "after": "LP unrestricted positive column0 = 0",
      "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 unrestricted positive column0，不以小數近似替代。",
          "en": "Recompute LP unrestricted positive column0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "18",
          "0"
        ],
        "output": "0",
        "label": "LP unrestricted positive column0"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "LP unrestricted negative column0 := mul(-18, 18)",
      "after": "LP unrestricted negative column0 = -324",
      "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 unrestricted negative column0，不以小數近似替代。",
          "en": "Recompute LP unrestricted negative column0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-18",
          "18"
        ],
        "output": "-324",
        "label": "LP unrestricted negative column0"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "LP unrestricted split column0 := sum(0, -324)",
      "after": "LP unrestricted split column0 = -324",
      "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 unrestricted split column0，不以小數近似替代。",
          "en": "Recompute LP unrestricted split column0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "-324"
        ],
        "output": "-324",
        "label": "LP unrestricted split column0"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "LP unrestricted ORIGINAL column1 := mul(-19, -18)",
      "after": "LP unrestricted ORIGINAL column1 = 342",
      "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 unrestricted ORIGINAL column1，不以小數近似替代。",
          "en": "Recompute LP unrestricted ORIGINAL column1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-19",
          "-18"
        ],
        "output": "342",
        "label": "LP unrestricted ORIGINAL column1"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "LP unrestricted positive column1 := mul(-19, 0)",
      "after": "LP unrestricted positive column1 = 0",
      "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 unrestricted positive column1，不以小數近似替代。",
          "en": "Recompute LP unrestricted positive column1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-19",
          "0"
        ],
        "output": "0",
        "label": "LP unrestricted positive column1"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "LP unrestricted negative column1 := mul(19, 18)",
      "after": "LP unrestricted negative column1 = 342",
      "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 unrestricted negative column1，不以小數近似替代。",
          "en": "Recompute LP unrestricted negative column1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "19",
          "18"
        ],
        "output": "342",
        "label": "LP unrestricted negative column1"
      }
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "LP unrestricted split column1 := sum(0, 342)",
      "after": "LP unrestricted split column1 = 342",
      "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 unrestricted split column1，不以小數近似替代。",
          "en": "Recompute LP unrestricted split column1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "342"
        ],
        "output": "342",
        "label": "LP unrestricted split column1"
      }
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "LP unrestricted ORIGINAL column2 := mul(20, -18)",
      "after": "LP unrestricted ORIGINAL column2 = -360",
      "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 unrestricted ORIGINAL column2，不以小數近似替代。",
          "en": "Recompute LP unrestricted ORIGINAL column2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "20",
          "-18"
        ],
        "output": "-360",
        "label": "LP unrestricted ORIGINAL column2"
      }
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "LP unrestricted positive column2 := mul(20, 0)",
      "after": "LP unrestricted positive column2 = 0",
      "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 unrestricted positive column2，不以小數近似替代。",
          "en": "Recompute LP unrestricted positive column2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "20",
          "0"
        ],
        "output": "0",
        "label": "LP unrestricted positive column2"
      }
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "LP unrestricted negative column2 := mul(-20, 18)",
      "after": "LP unrestricted negative column2 = -360",
      "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 unrestricted negative column2，不以小數近似替代。",
          "en": "Recompute LP unrestricted negative column2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-20",
          "18"
        ],
        "output": "-360",
        "label": "LP unrestricted negative column2"
      }
    },
    {
      "id": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "LP unrestricted split column2 := sum(0, -360)",
      "after": "LP unrestricted split column2 = -360",
      "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 unrestricted split column2，不以小數近似替代。",
          "en": "Recompute LP unrestricted split column2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "-360"
        ],
        "output": "-360",
        "label": "LP unrestricted split column2"
      }
    },
    {
      "id": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "LP unrestricted ORIGINAL objective contribution := mul(21, -18)",
      "after": "LP unrestricted ORIGINAL objective contribution = -378",
      "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 unrestricted ORIGINAL objective contribution，不以小數近似替代。",
          "en": "Recompute LP unrestricted ORIGINAL objective contribution with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "21",
          "-18"
        ],
        "output": "-378",
        "label": "LP unrestricted ORIGINAL objective contribution"
      }
    },
    {
      "id": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "LP unrestricted positive objective := mul(21, 0)",
      "after": "LP unrestricted positive objective = 0",
      "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 unrestricted positive objective，不以小數近似替代。",
          "en": "Recompute LP unrestricted positive objective with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "21",
          "0"
        ],
        "output": "0",
        "label": "LP unrestricted positive objective"
      }
    },
    {
      "id": "S017",
      "uses": [
        "G001",
        "S016"
      ],
      "before": "LP unrestricted negative objective := mul(-21, 18)",
      "after": "LP unrestricted negative objective = -378",
      "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 unrestricted negative objective，不以小數近似替代。",
          "en": "Recompute LP unrestricted negative objective with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-21",
          "18"
        ],
        "output": "-378",
        "label": "LP unrestricted negative objective"
      }
    },
    {
      "id": "S018",
      "uses": [
        "G001",
        "S017"
      ],
      "before": "LP unrestricted split objective sum := sum(0, -378)",
      "after": "LP unrestricted split objective sum = -378",
      "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 unrestricted split objective sum，不以小數近似替代。",
          "en": "Recompute LP unrestricted split objective sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "-378"
        ],
        "output": "-378",
        "label": "LP unrestricted split objective sum"
      }
    },
    {
      "id": "S019",
      "uses": [
        "G001",
        "S018"
      ],
      "before": "source roles = [[18, -19, 20], 21, -18, 0, 18]",
      "after": "derived = [-18, [-324, 342, -360], [-324, 342, -360], 是, -378, -378, 是]",
      "relation": "implies",
      "rule_id": "LP_nonnegative_split_preserves_contributions_NOT_all_pair_bijection",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正負部分均非負且v=plus−minus；每行a*v=a*plus+(−a)*minus，目標同理。反向投影合法，但對任意s>=0同加s仍表示同一v，故不是全體split pairs的一一對應。",
        "en": "Both parts are nonnegative and v=plus−minus.Each row satisfies a*v=a*plus+(−a)*minus,and so does the objective.Projection back is valid,but adding any s>=0 to both parts gives the same v,so this is not a bijection of all split pairs."
      },
      "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": [
      [
        "18",
        "-19",
        "20"
      ],
      "21",
      "-18",
      "0",
      "18",
      "-18",
      [
        "-324",
        "342",
        "-360"
      ],
      [
        "-324",
        "342",
        "-360"
      ],
      true,
      "-378",
      "-378",
      true,
      "x=x_plus-x_minus_with_both_nonnegative_preserves_column_and_objective_contribution"
    ]
  },
  "conclusion": {
    "canonical_value": "([18, -19, 20], 21, -18, 0, 18, -18, [-324, 342, -360], [-324, 342, -360], 是, -378, -378, 是, x=x_plus-x_minus_with_both_nonnegative_preserves_column_and_objective_contribution)",
    "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": 68,
      "original_line_sha256": "159eb0732f5d8df56a33c0376f2639fbb5e90f1754c8aa73320328b5a20a4588",
      "question_compact_sha256": "ed9e46ac5ca0e4eb8aa13ca9e5b85f5bdc7cc184ce9c167b04ac290aa129426e",
      "english_row_compact_sha256": "ce2b8f1bf78ef5e4b591bc631c6d27f649102b899bc7a81a92f3559284961459",
      "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": "1344bb4b5ac0dd722ff32cf8c2659a562dd0d33039e1e860b32304180da849b7",
    "original_candidate_line_sha256": "1344bb4b5ac0dd722ff32cf8c2659a562dd0d33039e1e860b32304180da849b7"
  },
  "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": 19,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "319bb4cae24d2f990869f754d992e572429bd6f2d730e5098c91c7df234e61e0",
      "arithmetic_steps": 17
    },
    "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."
  }
}