{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261010-advance-20261014-r02-real-number-0006",
  "pilot_number": "R0012-06806",
  "number": 6806,
  "family_id": "qf-density14-real-ordered-midpoint",
  "title": {
    "zh-Hant": "第 6806 題：有序有理端點的中點距離",
    "en": "Question 6806: Ordered Rational Midpoint and Endpoint Gaps"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261010-advance-20261014-r02-real-number-0006",
    "batch_id": "batch-20261010-advance-20261014-r02-real-number-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-real-number"
    ],
    "title_zh": "有序有理端點的中點距離",
    "stem_zh": "有理點嵌入實數的有限切片；a=-5/2≤b=6/3。依序求 (中點, 中點−a, b−中點)，用既約正分母分數。",
    "answer": {
      "kind": "expression",
      "value": "(-1/4, 9/4, 9/4)",
      "explanation_zh": "有序中點與兩側距離=(-1/4, 9/4, 9/4)。精確加總除二，再分別相減核兩側距離。非實數完備性或無理數分類證明，不用近似小數。"
    },
    "difficulty": {
      "band": "upper_secondary",
      "level": 1,
      "skills": [
        "有序有理端點的中點距離"
      ]
    },
    "variant": {
      "family_id": "qf-density14-real-ordered-midpoint",
      "seed": 202610142006,
      "parameters": {
        "a": [
          -5,
          2
        ],
        "b": [
          6,
          3
        ],
        "domain": "finite_rational_real_slice",
        "operation": "density14_real_ordered_midpoint",
        "semantic_template": "qf-density14-real-ordered-midpoint/[\"-5/2\",\"2/1\"]"
      }
    },
    "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": "2026-10-10",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261010-advance-20261014-r02-real-number-0006",
    "batch_id": "batch-20261010-advance-20261014-r02-real-number-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "94a54167303f819ad05d9abad8528f3984290aeb6789c1a3d35b718fe7147351",
      "family_id": "qf-density14-real-ordered-midpoint",
      "seed": 202610142006,
      "semantic_template": "qf-density14-real-ordered-midpoint/[\"-5/2\",\"2/1\"]"
    },
    "fields": {
      "title": "Ordered Rational Midpoint and Endpoint Gaps",
      "stem": "Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. a=-5/2≤b=6/3. Give (midpoint, midpoint−a, b−midpoint). Use exact reduced fractions with positive denominators, retaining /1 for integers.",
      "answer": {
        "kind": "expression",
        "value": "(-1/4, 9/4, 9/4)",
        "explanation": "Add and divide by two exactly, then subtract each endpoint separately; no irrational classification or real-completeness proof is claimed. The exact result, retaining the requested order, is (-1/4, 9/4, 9/4)."
      },
      "skills": [
        "Ordered Rational Midpoint and Endpoint Gaps"
      ],
      "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": "有理點嵌入實數的有限切片；a=-5/2≤b=6/3。依序求 (中點, 中點−a, b−中點)，用既約正分母分數。",
        "en": "Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. a=-5/2≤b=6/3. Give (midpoint, midpoint−a, b−midpoint). Use exact reduced fractions with positive denominators, retaining /1 for integers."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "explicit rational points embedded in the real line",
      "after": "exact finite order and distance data,not real completeness or irrational classification",
      "relation": "implies",
      "rule_id": "rational_real_slice_scope",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以精確有理值處理這個實數中的有限切片，不用近似小數，也不宣稱已證實數完備性或分類無理數。",
        "en": "Use exact rational values for this finite slice of the real line,without rounded decimals or claims about real completeness or classification of irrationals."
      },
      "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": "endpoint sum := sum(-5/2, 2)",
      "after": "endpoint sum = -1/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "兩個有序端點先完整相加。",
        "en": "Add the two ordered endpoints completely."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 endpoint sum，不以小數近似替代。",
          "en": "Recompute endpoint sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-5/2",
          "2"
        ],
        "output": "-1/2",
        "label": "endpoint sum"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "ordered midpoint := div(-1/2, 2)",
      "after": "ordered midpoint = -1/4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再除2得到精確中點，端點相等也可。",
        "en": "Divide by2 for the exact midpoint;equal endpoints are allowed."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 ordered midpoint，不以小數近似替代。",
          "en": "Recompute ordered midpoint with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "-1/2",
          "2"
        ],
        "output": "-1/4",
        "label": "ordered midpoint"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "midpoint minus left endpoint := sub(-1/4, -5/2)",
      "after": "midpoint minus left endpoint = 9/4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左側距離用中點減左端點。",
        "en": "The left distance is midpoint minus the left endpoint."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 midpoint minus left endpoint，不以小數近似替代。",
          "en": "Recompute midpoint minus left endpoint with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "-1/4",
          "-5/2"
        ],
        "output": "9/4",
        "label": "midpoint minus left endpoint"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "right endpoint minus midpoint := sub(2, -1/4)",
      "after": "right endpoint minus midpoint = 9/4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "右側距離用右端點減中點。",
        "en": "The right distance is the right endpoint minus midpoint."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 right endpoint minus midpoint，不以小數近似替代。",
          "en": "Recompute right endpoint minus midpoint with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "2",
          "-1/4"
        ],
        "output": "9/4",
        "label": "right endpoint minus midpoint"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "a=-5/2≤midpoint=-1/4≤b=2",
      "after": "two nonnegative distances equal 9/4",
      "relation": "implies",
      "rule_id": "ordered_midpoint_equidistance",
      "conditions": [],
      "explanation": {
        "zh-Hant": "端點有序使兩差是非負距離；a=b時兩距離0，不需排除退化區間。",
        "en": "Endpoint order makes both differences nonnegative distances;when a=b both are0,so no degenerate-interval exclusion is needed."
      },
      "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": [
      "-1/4",
      "9/4",
      "9/4"
    ]
  },
  "conclusion": {
    "canonical_value": "(-1/4, 9/4, 9/4)",
    "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": {
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    "question_binding": {
      "batch_id": "batch-20261010-advance-20261014-r02-real-number-foundations",
      "batch_path": "public/data/questions/batches/batch-20261010-advance-20261014-r02-real-number-foundations/questions.jsonl",
      "batch_sha256": "a21835c6da0df0833d78ad956806cd71881509cf26981f7af7c4dc492119e05c",
      "original_line": 6,
      "original_line_sha256": "cfadfd0c9678193043c9653ffebdb9a6f01d79aff89a79c771a3d856e8a24eab",
      "question_compact_sha256": "94a54167303f819ad05d9abad8528f3984290aeb6789c1a3d35b718fe7147351",
      "english_row_compact_sha256": "3a3face0895bba52c14cf7ea09d5c92a402437d3dff8a214916a3d6687d6652b",
      "english_manifest_path": "public/data/i18n/editions/batch-20261010-advance-20261014-r02-real-number-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261010-advance-20261014-r02-real-number-foundations.en.jsonl"
    },
    "mko_bindings": {
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        "english_sha256": "5dde700ff98e1ae0b35a5924253f8ad22515aa33c9a17deab8b7fa9a35e1ef5c",
        "original_path": "public/data/mko/real-number.json",
        "english_path": "public/data/i18n/editions/mko-real-number.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "c5c612ca91797c328f4489939ec6f7d885385e63ea21b334c09c421f17933d99",
    "original_candidate_sha256": "03b59ccbe98d57de37c5a58a10d05d4bdddb6b1a5a47fa037d71086a561cb0c3",
    "original_candidate_line_sha256": "03b59ccbe98d57de37c5a58a10d05d4bdddb6b1a5a47fa037d71086a561cb0c3"
  },
  "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-p13",
    "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": "90dbc494b2d34ef51f412dcff63cfcc1a2215d038bff71f38f5dded4c64faf00",
      "question_first_witness_sha256": "b72a3b8bcf10f584396e0dd5198c657b0934a71864e5a3fa6321d7a613aef9f9"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 6,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "6945885b1a65984841891eb3cb2d98938dfa5ab81b65a8cb92cc934f61047e5f",
      "arithmetic_steps": 4
    },
    "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."
  }
}