{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261010-advance-20261014-r02-real-number-0081",
  "pilot_number": "R0012-06881",
  "number": 6881,
  "family_id": "qf-density14-real-point-offset-distances",
  "title": {
    "zh-Hant": "第 6881 題：有理點的有向偏移與距離",
    "en": "Question 6881: Ordered Rational Offsets and Absolute Distances"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261010-advance-20261014-r02-real-number-0081",
    "batch_id": "batch-20261010-advance-20261014-r02-real-number-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-real-number"
    ],
    "title_zh": "有理點的有向偏移與距離",
    "stem_zh": "有理點嵌入實數的有限切片；origin=-7/2，有序points=[-35/6, -4/1, -7/2, -17/6]。依序列 (各point−origin, 各偏移絕對值)，保留四欄次序。",
    "answer": {
      "kind": "expression",
      "value": "([-7/3, -1/2, 0/1, 2/3], [7/3, 1/2, 0/1, 2/3])",
      "explanation_zh": "有理點有向偏移與距離=([-7/3, -1/2, 0/1, 2/3], [7/3, 1/2, 0/1, 2/3])。先精確相減保留方向，再取非負絕對值。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": [
        "有理點的有向偏移與距離"
      ]
    },
    "variant": {
      "family_id": "qf-density14-real-point-offset-distances",
      "seed": 202610142081,
      "parameters": {
        "origin": [
          -7,
          2
        ],
        "points": [
          [
            -35,
            6
          ],
          [
            -4,
            1
          ],
          [
            -7,
            2
          ],
          [
            -17,
            6
          ]
        ],
        "domain": "finite_rational_real_slice",
        "operation": "density14_real_point_offset_distances",
        "semantic_template": "qf-density14-real-point-offset-distances/[\"-7/3\",\"-1/2\",\"0/1\",\"2/3\"]"
      }
    },
    "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-0081",
    "batch_id": "batch-20261010-advance-20261014-r02-real-number-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "70f1a2e848e73f6c49ab449e9dc993aaf8c55e1be9d19a668f4d42d5a1162291",
      "family_id": "qf-density14-real-point-offset-distances",
      "seed": 202610142081,
      "semantic_template": "qf-density14-real-point-offset-distances/[\"-7/3\",\"-1/2\",\"0/1\",\"2/3\"]"
    },
    "fields": {
      "title": "Ordered Rational Offsets and Absolute Distances",
      "stem": "Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. origin=-7/2, ordered points=[-35/6, -4/1, -7/2, -17/6]. Give (each point−origin, each offset’s absolute value), retaining the four-field order.",
      "answer": {
        "kind": "expression",
        "value": "([-7/3, -1/2, 0/1, 2/3], [7/3, 1/2, 0/1, 2/3])",
        "explanation": "Subtract exactly to retain direction, then take the nonnegative absolute value; directed offsets and distances are separate. The exact result, retaining the requested order, is ([-7/3, -1/2, 0/1, 2/3], [7/3, 1/2, 0/1, 2/3])."
      },
      "skills": [
        "Ordered Rational Offsets and Absolute Distances"
      ],
      "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；origin=-7/2，有序points=[-35/6, -4/1, -7/2, -17/6]。依序列 (各point−origin, 各偏移絕對值)，保留四欄次序。",
        "en": "Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. origin=-7/2, ordered points=[-35/6, -4/1, -7/2, -17/6]. Give (each point−origin, each offset’s absolute value), retaining the four-field order."
      }
    }
  ],
  "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": "point0 signed offset := sub(-35/6, -7/2)",
      "after": "point0 signed offset = -7/3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留point−origin的有向差，不先取絕對值。",
        "en": "Retain the directed difference point−origin before taking an absolute value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 point0 signed offset，不以小數近似替代。",
          "en": "Recompute point0 signed offset with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "-35/6",
          "-7/2"
        ],
        "output": "-7/3",
        "label": "point0 signed offset"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "point0 opposite negative offset := mul(-1, -7/3)",
      "after": "point0 opposite negative offset = 7/3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "負偏移乘−1才成非負距離。",
        "en": "Multiply a negative offset by−1 to obtain nonnegative distance."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 point0 opposite negative offset，不以小數近似替代。",
          "en": "Recompute point0 opposite negative offset with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-1",
          "-7/3"
        ],
        "output": "7/3",
        "label": "point0 opposite negative offset"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "point=-35/6;origin=-7/2;offset=-7/3",
      "after": "absolute distance=7/3",
      "relation": "implies",
      "rule_id": "signed_point_offset_absolute_distance",
      "conditions": [],
      "explanation": {
        "zh-Hant": "非負偏移保持原值，負偏移取相反數；距離非負但原有方向資訊仍在第一列表。",
        "en": "Keep a nonnegative offset and negate a negative one;distance is nonnegative while the first list retains direction."
      },
      "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": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "point1 signed offset := sub(-4, -7/2)",
      "after": "point1 signed offset = -1/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留point−origin的有向差，不先取絕對值。",
        "en": "Retain the directed difference point−origin before taking an absolute value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 point1 signed offset，不以小數近似替代。",
          "en": "Recompute point1 signed offset with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "-4",
          "-7/2"
        ],
        "output": "-1/2",
        "label": "point1 signed offset"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "point1 opposite negative offset := mul(-1, -1/2)",
      "after": "point1 opposite negative offset = 1/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "負偏移乘−1才成非負距離。",
        "en": "Multiply a negative offset by−1 to obtain nonnegative distance."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 point1 opposite negative offset，不以小數近似替代。",
          "en": "Recompute point1 opposite negative offset with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-1",
          "-1/2"
        ],
        "output": "1/2",
        "label": "point1 opposite negative offset"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "point=-4;origin=-7/2;offset=-1/2",
      "after": "absolute distance=1/2",
      "relation": "implies",
      "rule_id": "signed_point_offset_absolute_distance",
      "conditions": [],
      "explanation": {
        "zh-Hant": "非負偏移保持原值，負偏移取相反數；距離非負但原有方向資訊仍在第一列表。",
        "en": "Keep a nonnegative offset and negate a negative one;distance is nonnegative while the first list retains direction."
      },
      "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": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "point2 signed offset := sub(-7/2, -7/2)",
      "after": "point2 signed offset = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留point−origin的有向差，不先取絕對值。",
        "en": "Retain the directed difference point−origin before taking an absolute value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 point2 signed offset，不以小數近似替代。",
          "en": "Recompute point2 signed offset with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "-7/2",
          "-7/2"
        ],
        "output": "0",
        "label": "point2 signed offset"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "point=-7/2;origin=-7/2;offset=0",
      "after": "absolute distance=0/1",
      "relation": "implies",
      "rule_id": "signed_point_offset_absolute_distance",
      "conditions": [],
      "explanation": {
        "zh-Hant": "非負偏移保持原值，負偏移取相反數；距離非負但原有方向資訊仍在第一列表。",
        "en": "Keep a nonnegative offset and negate a negative one;distance is nonnegative while the first list retains direction."
      },
      "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": "point3 signed offset := sub(-17/6, -7/2)",
      "after": "point3 signed offset = 2/3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留point−origin的有向差，不先取絕對值。",
        "en": "Retain the directed difference point−origin before taking an absolute value."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 point3 signed offset，不以小數近似替代。",
          "en": "Recompute point3 signed offset with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "-17/6",
          "-7/2"
        ],
        "output": "2/3",
        "label": "point3 signed offset"
      }
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "point=-17/6;origin=-7/2;offset=2/3",
      "after": "absolute distance=2/3",
      "relation": "implies",
      "rule_id": "signed_point_offset_absolute_distance",
      "conditions": [],
      "explanation": {
        "zh-Hant": "非負偏移保持原值，負偏移取相反數；距離非負但原有方向資訊仍在第一列表。",
        "en": "Keep a nonnegative offset and negate a negative one;distance is nonnegative while the first list retains direction."
      },
      "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": [
      [
        "-7/3",
        "-1/2",
        "0",
        "2/3"
      ],
      [
        "7/3",
        "1/2",
        "0",
        "2/3"
      ]
    ]
  },
  "conclusion": {
    "canonical_value": "([-7/3, -1/2, 0/1, 2/3], [7/3, 1/2, 0/1, 2/3])",
    "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-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",
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      "original_line_sha256": "a894e134b47239c1cd09c4821c242d17d8de00537e27ebd9e8846892fe9e9ea4",
      "question_compact_sha256": "70f1a2e848e73f6c49ab449e9dc993aaf8c55e1be9d19a668f4d42d5a1162291",
      "english_row_compact_sha256": "1a267f64d17a7d92df4b036f4bc9e070f8e92da967ec363dc8d79fbd65a2413c",
      "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": {
      "mko-real-number": {
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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": "1b36294dc4dbf7a7523894af3acf5ffb524fa8c25b3dd283b2c988c0ed5142ef",
    "original_candidate_line_sha256": "1b36294dc4dbf7a7523894af3acf5ffb524fa8c25b3dd283b2c988c0ed5142ef"
  },
  "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": 11,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "1696acae7e8a58d1fe91370ddc7cefd7d425ada81df2241d18add6b6b10b1e7d",
      "arithmetic_steps": 6
    },
    "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."
  }
}