{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261012-advance-20261017-r04-order-relation-0011",
  "pilot_number": "R0015-08286",
  "number": 8286,
  "family_id": "qf-density17-order-three-rational-sort",
  "title": {
    "zh-Hant": "第 8286 題：三分數的穩定有序索引",
    "en": "Question 8286: Stable Rational Sort with Original Index Witnesses"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261012-advance-20261017-r04-order-relation-0011",
    "batch_id": "batch-20261012-advance-20261017-r04-order-relation-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-order-relation"
    ],
    "title_zh": "三分數的穩定有序索引",
    "stem_zh": "R通常全序，原分母為正。原序values=[(-2)/(3), (1)/(2), (-1)/(4)]，索引從0起。答 (原序既約值, 由小到大的原索引, 已排序既約值, 相鄰兩比較符號)。同值保留原索引先後。",
    "answer": {
      "kind": "expression",
      "value": "([-2/3, 1/2, -1/4], [0, 2, 1], [-2/3, -1/4, 1/2], [<, <])",
      "explanation_zh": "離散結構核對=([-2/3, 1/2, -1/4], [0, 2, 1], [-2/3, -1/4, 1/2], [<, <])；正分母交叉乘積比較，穩定插入索引以保留同值先後。；有限精確核對不是一般形式證明；閉區間含端點；乘負反向乘零合併；不外推任意偏序或複數。"
    },
    "difficulty": {
      "band": "lower_secondary",
      "level": 2,
      "skills": [
        "三分數的穩定有序索引"
      ]
    },
    "variant": {
      "family_id": "qf-density17-order-three-rational-sort",
      "seed": 202610174011,
      "parameters": {
        "values": [
          [
            -2,
            3
          ],
          [
            1,
            2
          ],
          [
            -1,
            4
          ]
        ],
        "domain": "R_positive_denominator_order",
        "operation": "order-three-rational-sort",
        "semantic_template": "qf-density17-order-three-rational-sort/{\"values\":[\"-2/3\",\"1/2\",\"-1/4\"]}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_integer_rational_arithmetic",
        "producer_finite_enumeration",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-12",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261012-advance-20261017-r04-order-relation-0011",
    "batch_id": "batch-20261012-advance-20261017-r04-order-relation-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "bd7a2111c7036fa41f0cc0c1e484b982c0c4c07c3830c663d7e2b893d8a71606",
      "family_id": "qf-density17-order-three-rational-sort",
      "seed": 202610174011,
      "semantic_template": "qf-density17-order-three-rational-sort/{\"values\":[\"-2/3\",\"1/2\",\"-1/4\"]}"
    },
    "fields": {
      "title": "Stable Rational Sort with Original Index Witnesses",
      "stem": "Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Ordered inputs=[-2/3, 1/2, -1/4]. Give (normalized inputs, stable sorted original indices, sorted values, adjacent comparison symbols). Equal values retain their original index order.",
      "answer": {
        "kind": "expression",
        "value": "([-2/3, 1/2, -1/4], [0, 2, 1], [-2/3, -1/4, 1/2], [<, <])",
        "explanation": "Compare exact rational cross-products and use original indices to break equal-value ties. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-2/3, 1/2, -1/4], [0, 2, 1], [-2/3, -1/4, 1/2], [<, <])."
      },
      "skills": [
        "Stable Rational Sort with Original Index Witnesses"
      ],
      "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": "R通常全序，原分母為正。原序values=[(-2)/(3), (1)/(2), (-1)/(4)]，索引從0起。答 (原序既約值, 由小到大的原索引, 已排序既約值, 相鄰兩比較符號)。同值保留原索引先後。",
        "en": "Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Ordered inputs=[-2/3, 1/2, -1/4]. Give (normalized inputs, stable sorted original indices, sorted values, adjacent comparison symbols). Equal values retain their original index order."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "usual total order on R;all original denominators positive",
      "after": "closed endpoints included;negative scaling reverses;zero collapses",
      "relation": "implies",
      "rule_id": "real_rational_order_scope",
      "conditions": [],
      "explanation": {
        "zh-Hant": "此題是通常實數全序的精確有理切片，不外推複數或任意偏序；原分母必正，不能先正規化負分母來略過原規格。",
        "en": "These exact rationals use the usual real total order,not complex numbers or an arbitrary partial order.Original denominators must be positive;normalizing a negative input denominator does not waive that contract."
      },
      "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": "original pair0,1 left cross := mul(-2, 2)",
      "after": "original pair0,1 left cross = -4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步指定順序精確代入原數據，分母非零才允許除法。",
        "en": "Substitute the source data exactly in the stated operand order;division requires a nonzero denominator."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 original pair0,1 left cross，不以小數近似替代。",
          "en": "Recompute original pair0,1 left cross with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-2",
          "2"
        ],
        "output": "-4",
        "label": "original pair0,1 left cross"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "original pair0,1 right cross := mul(1, 3)",
      "after": "original pair0,1 right cross = 3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步指定順序精確代入原數據，分母非零才允許除法。",
        "en": "Substitute the source data exactly in the stated operand order;division requires a nonzero denominator."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 original pair0,1 right cross，不以小數近似替代。",
          "en": "Recompute original pair0,1 right cross with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "3"
        ],
        "output": "3",
        "label": "original pair0,1 right cross"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "original pair0,1 cross residual := sub(-4, 3)",
      "after": "original pair0,1 cross residual = -7",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步指定順序精確代入原數據，分母非零才允許除法。",
        "en": "Substitute the source data exactly in the stated operand order;division requires a nonzero denominator."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 original pair0,1 cross residual，不以小數近似替代。",
          "en": "Recompute original pair0,1 cross residual with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "-4",
          "3"
        ],
        "output": "-7",
        "label": "original pair0,1 cross residual"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "original pair0,1:fractions=[-2/3, 1/2]",
      "after": "cross residual=-7;comparison=<",
      "relation": "implies",
      "rule_id": "rational_positive_denominator_comparison",
      "conditions": [],
      "explanation": {
        "zh-Hant": "既約分母皆正，交乘差的符號與兩數大小一致；保留等號，同值排序另外依原索引打破平手。",
        "en": "Reduced denominators are positive,so the cross-difference sign determines order.Retain equality and use original indices separately to break equal-value sorting ties."
      },
      "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": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "original pair0,2 left cross := mul(-2, 4)",
      "after": "original pair0,2 left cross = -8",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步指定順序精確代入原數據，分母非零才允許除法。",
        "en": "Substitute the source data exactly in the stated operand order;division requires a nonzero denominator."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 original pair0,2 left cross，不以小數近似替代。",
          "en": "Recompute original pair0,2 left cross with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-2",
          "4"
        ],
        "output": "-8",
        "label": "original pair0,2 left cross"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "original pair0,2 right cross := mul(-1, 3)",
      "after": "original pair0,2 right cross = -3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步指定順序精確代入原數據，分母非零才允許除法。",
        "en": "Substitute the source data exactly in the stated operand order;division requires a nonzero denominator."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 original pair0,2 right cross，不以小數近似替代。",
          "en": "Recompute original pair0,2 right cross with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-1",
          "3"
        ],
        "output": "-3",
        "label": "original pair0,2 right cross"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "original pair0,2 cross residual := sub(-8, -3)",
      "after": "original pair0,2 cross residual = -5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步指定順序精確代入原數據，分母非零才允許除法。",
        "en": "Substitute the source data exactly in the stated operand order;division requires a nonzero denominator."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 original pair0,2 cross residual，不以小數近似替代。",
          "en": "Recompute original pair0,2 cross residual with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "-8",
          "-3"
        ],
        "output": "-5",
        "label": "original pair0,2 cross residual"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "original pair0,2:fractions=[-2/3, -1/4]",
      "after": "cross residual=-5;comparison=<",
      "relation": "implies",
      "rule_id": "rational_positive_denominator_comparison",
      "conditions": [],
      "explanation": {
        "zh-Hant": "既約分母皆正，交乘差的符號與兩數大小一致；保留等號，同值排序另外依原索引打破平手。",
        "en": "Reduced denominators are positive,so the cross-difference sign determines order.Retain equality and use original indices separately to break equal-value sorting ties."
      },
      "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": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "original pair1,2 left cross := mul(1, 4)",
      "after": "original pair1,2 left cross = 4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步指定順序精確代入原數據，分母非零才允許除法。",
        "en": "Substitute the source data exactly in the stated operand order;division requires a nonzero denominator."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 original pair1,2 left cross，不以小數近似替代。",
          "en": "Recompute original pair1,2 left cross with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "4"
        ],
        "output": "4",
        "label": "original pair1,2 left cross"
      }
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "original pair1,2 right cross := mul(-1, 2)",
      "after": "original pair1,2 right cross = -2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步指定順序精確代入原數據，分母非零才允許除法。",
        "en": "Substitute the source data exactly in the stated operand order;division requires a nonzero denominator."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 original pair1,2 right cross，不以小數近似替代。",
          "en": "Recompute original pair1,2 right cross with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-1",
          "2"
        ],
        "output": "-2",
        "label": "original pair1,2 right cross"
      }
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "original pair1,2 cross residual := sub(4, -2)",
      "after": "original pair1,2 cross residual = 6",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步指定順序精確代入原數據，分母非零才允許除法。",
        "en": "Substitute the source data exactly in the stated operand order;division requires a nonzero denominator."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 original pair1,2 cross residual，不以小數近似替代。",
          "en": "Recompute original pair1,2 cross residual with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "4",
          "-2"
        ],
        "output": "6",
        "label": "original pair1,2 cross residual"
      }
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "original pair1,2:fractions=[1/2, -1/4]",
      "after": "cross residual=6;comparison=>",
      "relation": "implies",
      "rule_id": "rational_positive_denominator_comparison",
      "conditions": [],
      "explanation": {
        "zh-Hant": "既約分母皆正，交乘差的符號與兩數大小一致；保留等號，同值排序另外依原索引打破平手。",
        "en": "Reduced denominators are positive,so the cross-difference sign determines order.Retain equality and use original indices separately to break equal-value sorting ties."
      },
      "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": "sorted adjacent0 left cross := mul(-2, 4)",
      "after": "sorted adjacent0 left cross = -8",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步指定順序精確代入原數據，分母非零才允許除法。",
        "en": "Substitute the source data exactly in the stated operand order;division requires a nonzero denominator."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 sorted adjacent0 left cross，不以小數近似替代。",
          "en": "Recompute sorted adjacent0 left cross with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-2",
          "4"
        ],
        "output": "-8",
        "label": "sorted adjacent0 left cross"
      }
    },
    {
      "id": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "sorted adjacent0 right cross := mul(-1, 3)",
      "after": "sorted adjacent0 right cross = -3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步指定順序精確代入原數據，分母非零才允許除法。",
        "en": "Substitute the source data exactly in the stated operand order;division requires a nonzero denominator."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 sorted adjacent0 right cross，不以小數近似替代。",
          "en": "Recompute sorted adjacent0 right cross with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-1",
          "3"
        ],
        "output": "-3",
        "label": "sorted adjacent0 right cross"
      }
    },
    {
      "id": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "sorted adjacent0 cross residual := sub(-8, -3)",
      "after": "sorted adjacent0 cross residual = -5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步指定順序精確代入原數據，分母非零才允許除法。",
        "en": "Substitute the source data exactly in the stated operand order;division requires a nonzero denominator."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 sorted adjacent0 cross residual，不以小數近似替代。",
          "en": "Recompute sorted adjacent0 cross residual with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "-8",
          "-3"
        ],
        "output": "-5",
        "label": "sorted adjacent0 cross residual"
      }
    },
    {
      "id": "S017",
      "uses": [
        "G001",
        "S016"
      ],
      "before": "sorted adjacent0:fractions=[-2/3, -1/4]",
      "after": "cross residual=-5;comparison=<",
      "relation": "implies",
      "rule_id": "rational_positive_denominator_comparison",
      "conditions": [],
      "explanation": {
        "zh-Hant": "既約分母皆正，交乘差的符號與兩數大小一致；保留等號，同值排序另外依原索引打破平手。",
        "en": "Reduced denominators are positive,so the cross-difference sign determines order.Retain equality and use original indices separately to break equal-value sorting ties."
      },
      "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": "S018",
      "uses": [
        "G001",
        "S017"
      ],
      "before": "sorted adjacent1 left cross := mul(-1, 2)",
      "after": "sorted adjacent1 left cross = -2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步指定順序精確代入原數據，分母非零才允許除法。",
        "en": "Substitute the source data exactly in the stated operand order;division requires a nonzero denominator."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 sorted adjacent1 left cross，不以小數近似替代。",
          "en": "Recompute sorted adjacent1 left cross with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-1",
          "2"
        ],
        "output": "-2",
        "label": "sorted adjacent1 left cross"
      }
    },
    {
      "id": "S019",
      "uses": [
        "G001",
        "S018"
      ],
      "before": "sorted adjacent1 right cross := mul(1, 4)",
      "after": "sorted adjacent1 right cross = 4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步指定順序精確代入原數據，分母非零才允許除法。",
        "en": "Substitute the source data exactly in the stated operand order;division requires a nonzero denominator."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 sorted adjacent1 right cross，不以小數近似替代。",
          "en": "Recompute sorted adjacent1 right cross with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "4"
        ],
        "output": "4",
        "label": "sorted adjacent1 right cross"
      }
    },
    {
      "id": "S020",
      "uses": [
        "G001",
        "S019"
      ],
      "before": "sorted adjacent1 cross residual := sub(-2, 4)",
      "after": "sorted adjacent1 cross residual = -6",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步指定順序精確代入原數據，分母非零才允許除法。",
        "en": "Substitute the source data exactly in the stated operand order;division requires a nonzero denominator."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 sorted adjacent1 cross residual，不以小數近似替代。",
          "en": "Recompute sorted adjacent1 cross residual with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "-2",
          "4"
        ],
        "output": "-6",
        "label": "sorted adjacent1 cross residual"
      }
    },
    {
      "id": "S021",
      "uses": [
        "G001",
        "S020"
      ],
      "before": "sorted adjacent1:fractions=[-1/4, 1/2]",
      "after": "cross residual=-6;comparison=<",
      "relation": "implies",
      "rule_id": "rational_positive_denominator_comparison",
      "conditions": [],
      "explanation": {
        "zh-Hant": "既約分母皆正，交乘差的符號與兩數大小一致；保留等號，同值排序另外依原索引打破平手。",
        "en": "Reduced denominators are positive,so the cross-difference sign determines order.Retain equality and use original indices separately to break equal-value sorting ties."
      },
      "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": "S022",
      "uses": [
        "G001",
        "S021"
      ],
      "before": "normalized original values=[-2/3, 1/2, -1/4]",
      "after": "stable original indices=[0, 2, 1];sorted values=[-2/3, -1/4, 1/2]",
      "relation": "implies",
      "rule_id": "stable_rational_original_index_sort",
      "conditions": [],
      "explanation": {
        "zh-Hant": "比較值相同時保留原索引順序；值可重複，不能去重成集合。",
        "en": "Preserve original index order when values tie.Repeated values remain separate entries rather than being deduplicated as a set."
      },
      "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": [
      [
        "-2/3",
        "1/2",
        "-1/4"
      ],
      [
        0,
        2,
        1
      ],
      [
        "-2/3",
        "-1/4",
        "1/2"
      ],
      [
        "<",
        "<"
      ]
    ]
  },
  "conclusion": {
    "canonical_value": "([-2/3, 1/2, -1/4], [0, 2, 1], [-2/3, -1/4, 1/2], [<, <])",
    "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-20261012-advance-20261017-r04-order-relation-foundations",
      "batch_path": "public/data/questions/batches/batch-20261012-advance-20261017-r04-order-relation-foundations/questions.jsonl",
      "batch_sha256": "6bee78c2e753258a82042d9051a2179c845945ab84af56cbb9e83a8e573cf599",
      "original_line": 11,
      "original_line_sha256": "fa43286ec039e785cddf4d144db06e30f14fa54ea8e4d7947c160179a688b0ab",
      "question_compact_sha256": "bd7a2111c7036fa41f0cc0c1e484b982c0c4c07c3830c663d7e2b893d8a71606",
      "english_row_compact_sha256": "ab4da6dd4c5fdc2cc4c675f44c4e66bfc0e783c2d295047325d823aaf64f01a2",
      "english_manifest_path": "public/data/i18n/editions/batch-20261012-advance-20261017-r04-order-relation-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261012-advance-20261017-r04-order-relation-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-order-relation": {
        "sha256": "95554189da3a451f8a69edd139e154fd9254524119837ea1643487a5646a2dc3",
        "english_sha256": "3bb53f0ce66755124a9e09f8d185cdebfbb4e6f8cc93a7733a5c08ceba6bdcd6",
        "original_path": "public/data/mko/order-relation.json",
        "english_path": "public/data/i18n/editions/mko-order-relation.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "ca1078dc467a0f3769865919aadc46c87712d098f1dfbde90f787471e22bcfec",
    "original_candidate_sha256": "cebe7d666680cbec1f146f99c787b5064f2536b60672d3e9ab29403382c6add8",
    "original_candidate_line_sha256": "cebe7d666680cbec1f146f99c787b5064f2536b60672d3e9ab29403382c6add8"
  },
  "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-20261026-005-p16",
    "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": "6ecad9d909199221a07e923f87425e80b3bdec3eed46fe5910d04fca694cae18",
      "question_first_witness_sha256": "e8f00458977a09493b5f9f05ab9e6feec72debaf4c7d1c609d1fc95900a35a1b"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 22,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "df25396cd94e34a9a004f6cdcaf29af47e751c4ddc78d124a2f45312f0e5a814",
      "arithmetic_steps": 15
    },
    "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."
  }
}