{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261012-advance-20261017-r04-order-relation-0041",
  "pilot_number": "R0015-08216",
  "number": 8216,
  "family_id": "qf-density17-order-interval-intersection",
  "title": {
    "zh-Hant": "第 8216 題：兩個閉區間的交集門檻",
    "en": "Question 8216: Closed-Interval Intersection Including Touching Endpoints"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261012-advance-20261017-r04-order-relation-0041",
    "batch_id": "batch-20261012-advance-20261017-r04-order-relation-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-order-relation"
    ],
    "title_zh": "兩個閉區間的交集門檻",
    "stem_zh": "R通常全序，原分母為正。閉區間I1=[(3)/(2), (9)/(2)]，I2=[(6)/(2), (12)/(2)]。答 (I1兩界, I2兩界, 最大下界, 最小上界, 交集非空資格, 交集兩界或空集)。",
    "answer": {
      "kind": "expression",
      "value": "([3/2, 9/2], [3/1, 6/1], 3/1, 9/2, 是, [3/1, 9/2])",
      "explanation_zh": "離散結構核對=([3/2, 9/2], [3/1, 6/1], 3/1, 9/2, 是, [3/1, 9/2])；比較兩下界取大、兩上界取小；下界等於上界仍含該點。；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": [
        "兩個閉區間的交集門檻"
      ]
    },
    "variant": {
      "family_id": "qf-density17-order-interval-intersection",
      "seed": 202610174041,
      "parameters": {
        "l1": [
          3,
          2
        ],
        "u1": [
          9,
          2
        ],
        "l2": [
          6,
          2
        ],
        "u2": [
          12,
          2
        ],
        "domain": "R_positive_denominator_order",
        "operation": "order-interval-intersection",
        "semantic_template": "qf-density17-order-interval-intersection/{\"l1\":\"3/2\",\"l2\":\"3/1\",\"u1\":\"9/2\",\"u2\":\"6/1\"}"
      }
    },
    "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-0041",
    "batch_id": "batch-20261012-advance-20261017-r04-order-relation-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "48eb2c36bba198604a7049017952935d858c2f40f02ff00a09da21cf5504d92f",
      "family_id": "qf-density17-order-interval-intersection",
      "seed": 202610174041,
      "semantic_template": "qf-density17-order-interval-intersection/{\"l1\":\"3/2\",\"l2\":\"3/1\",\"u1\":\"9/2\",\"u2\":\"6/1\"}"
    },
    "fields": {
      "title": "Closed-Interval Intersection Including Touching Endpoints",
      "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. Intervals are [3/2, 9/2] and [6/2, 12/2]. Give (first endpoints, second endpoints, maximum lower bound, minimum upper bound, nonempty?, intersection endpoints or empty-set).",
      "answer": {
        "kind": "expression",
        "value": "([3/2, 9/2], [3/1, 6/1], 3/1, 9/2, Yes, [3/1, 9/2])",
        "explanation": "Use inclusive endpoint comparisons; equality of the two inner bounds gives a singleton, not an empty intersection. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3/2, 9/2], [3/1, 6/1], 3/1, 9/2, Yes, [3/1, 9/2])."
      },
      "skills": [
        "Closed-Interval Intersection Including Touching Endpoints"
      ],
      "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。閉區間I1=[(3)/(2), (9)/(2)]，I2=[(6)/(2), (12)/(2)]。答 (I1兩界, I2兩界, 最大下界, 最小上界, 交集非空資格, 交集兩界或空集)。",
        "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. Intervals are [3/2, 9/2] and [6/2, 12/2]. Give (first endpoints, second endpoints, maximum lower bound, minimum upper bound, nonempty?, intersection endpoints or empty-set)."
      }
    }
  ],
  "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": "lower bounds left cross := mul(3, 1)",
      "after": "lower bounds left 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 lower bounds left cross，不以小數近似替代。",
          "en": "Recompute lower bounds left cross with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "3",
          "1"
        ],
        "output": "3",
        "label": "lower bounds left cross"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "lower bounds right cross := mul(3, 2)",
      "after": "lower bounds right cross = 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 lower bounds right cross，不以小數近似替代。",
          "en": "Recompute lower bounds right cross with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "3",
          "2"
        ],
        "output": "6",
        "label": "lower bounds right cross"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "lower bounds cross residual := sub(3, 6)",
      "after": "lower bounds cross residual = -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 lower bounds cross residual，不以小數近似替代。",
          "en": "Recompute lower bounds cross residual with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "3",
          "6"
        ],
        "output": "-3",
        "label": "lower bounds cross residual"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "lower bounds:fractions=[3/2, 3/1]",
      "after": "cross residual=-3;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": "upper bounds left cross := mul(9, 1)",
      "after": "upper bounds left cross = 9",
      "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 upper bounds left cross，不以小數近似替代。",
          "en": "Recompute upper bounds left cross with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "9",
          "1"
        ],
        "output": "9",
        "label": "upper bounds left cross"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "upper bounds right cross := mul(6, 2)",
      "after": "upper bounds right cross = 12",
      "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 upper bounds right cross，不以小數近似替代。",
          "en": "Recompute upper bounds right cross with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "6",
          "2"
        ],
        "output": "12",
        "label": "upper bounds right cross"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "upper bounds cross residual := sub(9, 12)",
      "after": "upper bounds cross residual = -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 upper bounds cross residual，不以小數近似替代。",
          "en": "Recompute upper bounds cross residual with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "9",
          "12"
        ],
        "output": "-3",
        "label": "upper bounds cross residual"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "upper bounds:fractions=[9/2, 6/1]",
      "after": "cross residual=-3;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": "intersection inner bounds left cross := mul(3, 2)",
      "after": "intersection inner bounds left cross = 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 intersection inner bounds left cross，不以小數近似替代。",
          "en": "Recompute intersection inner bounds left cross with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "3",
          "2"
        ],
        "output": "6",
        "label": "intersection inner bounds left cross"
      }
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "intersection inner bounds right cross := mul(9, 1)",
      "after": "intersection inner bounds right cross = 9",
      "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 intersection inner bounds right cross，不以小數近似替代。",
          "en": "Recompute intersection inner bounds right cross with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "9",
          "1"
        ],
        "output": "9",
        "label": "intersection inner bounds right cross"
      }
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "intersection inner bounds cross residual := sub(6, 9)",
      "after": "intersection inner bounds cross residual = -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 intersection inner bounds cross residual，不以小數近似替代。",
          "en": "Recompute intersection inner bounds cross residual with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "6",
          "9"
        ],
        "output": "-3",
        "label": "intersection inner bounds cross residual"
      }
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "intersection inner bounds:fractions=[3/1, 9/2]",
      "after": "cross residual=-3;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": "intervals=[[3/2, 9/2], [3/1, 6/1]]",
      "after": "max lower=3/1;min upper=9/2;nonempty=是",
      "relation": "implies",
      "rule_id": "closed_interval_intersection_guard",
      "conditions": [],
      "explanation": {
        "zh-Hant": "同時滿足兩個下界等於取較大下界，同時滿足兩個上界等於取較小上界；兩內界相等仍含那一點。",
        "en": "Satisfying both lower bounds means using their maximum;both upper bounds means their minimum.Equal inner bounds still include that single point."
      },
      "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": [
      [
        "3/2",
        "9/2"
      ],
      [
        "3/1",
        "6/1"
      ],
      "3/1",
      "9/2",
      true,
      [
        "3/1",
        "9/2"
      ]
    ]
  },
  "conclusion": {
    "canonical_value": "([3/2, 9/2], [3/1, 6/1], 3/1, 9/2, 是, [3/1, 9/2])",
    "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": 41,
      "original_line_sha256": "a17a82bf2cb0d79dd01f203cc0838223bb706da114ac5e76f0653dc957574dcc",
      "question_compact_sha256": "48eb2c36bba198604a7049017952935d858c2f40f02ff00a09da21cf5504d92f",
      "english_row_compact_sha256": "7480954610d21059a70e2e5f31a8f1dc57dd1863bda10cb92c170a3c77aca21e",
      "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"
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    "mko_bindings": {
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        "original_path": "public/data/mko/order-relation.json",
        "english_path": "public/data/i18n/editions/mko-order-relation.en.json"
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    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "ca1078dc467a0f3769865919aadc46c87712d098f1dfbde90f787471e22bcfec",
    "original_candidate_sha256": "5911e808a718bfcbb4d60ed14d20062960c9915483cba50035bf45c7616265da",
    "original_candidate_line_sha256": "5911e808a718bfcbb4d60ed14d20062960c9915483cba50035bf45c7616265da"
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      "source_question_state": "candidate",
      "independent_AI_or_human_review": "NOT_PERFORMED",
      "formalization": "not_formalized",
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    },
    "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": {
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      "question_first_witness_sha256": "e8f00458977a09493b5f9f05ab9e6feec72debaf4c7d1c609d1fc95900a35a1b"
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    "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": "83584ed8801aaf464c242ff8e84c7f9d334bbb1c064daa7fab441828f07003a0",
      "arithmetic_steps": 9
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    "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."
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}