{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261013-advance-20261020-r03-numerical-approximation-error-0100",
  "pilot_number": "R0002-01650",
  "number": 1650,
  "family_id": "qf-app20-error-error-interval",
  "title": {
    "zh-Hant": "第 1650 題：精確有理誤差",
    "en": "Question 1650: Inclusive Error Interval and a Candidate Boundary"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261013-advance-20261020-r03-numerical-approximation-error-0100",
    "batch_id": "batch-20261013-advance-20261020-r03-numerical-approximation-error-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-numerical-approximation-error"
    ],
    "title_zh": "精確有理誤差",
    "stem_zh": "精確有理數。操作=error-error-interval；資料={\"reference\":[7,1],\"tolerance\":[1,5],\"candidate\":[49,5]}。請依 [reference,tolerance,lower,upper,candidate,in_interval] 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(7, 1/5, 34/5, 36/5, 49/5, 否)",
      "explanation_zh": "應用基礎核對=(7, 1/5, 34/5, 36/5, 49/5, 否)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；relative error要求reference非零；不混淆截斷誤差與捨入誤差。"
    },
    "difficulty": {
      "band": "upper_secondary",
      "level": 3,
      "skills": [
        "精確有理誤差"
      ]
    },
    "variant": {
      "family_id": "qf-app20-error-error-interval",
      "seed": 202610203100,
      "parameters": {
        "reference": [
          7,
          1
        ],
        "tolerance": [
          1,
          5
        ],
        "candidate": [
          49,
          5
        ],
        "domain": "Exact_rational_error",
        "operation": "error-error-interval",
        "semantic_template": "qf-app20-error-error-interval/{\"candidate\":[49,5],\"reference\":[7,1],\"tolerance\":[1,5]}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_finite_model",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-13",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261013-advance-20261020-r03-numerical-approximation-error-0100",
    "batch_id": "batch-20261013-advance-20261020-r03-numerical-approximation-error-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "81220a1da9be1a111265c6c1f4ba5ff440bec554db5250a46602da776b59fd96",
      "family_id": "qf-app20-error-error-interval",
      "seed": 202610203100,
      "semantic_template": "qf-app20-error-error-interval/{\"candidate\":[49,5],\"reference\":[7,1],\"tolerance\":[1,5]}"
    },
    "fields": {
      "title": "Inclusive Error Interval and a Candidate Boundary",
      "stem": "Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. reference=7/1, tolerance=1/5≥0, candidate=49/5. Give (reference, tolerance, reference−tolerance, reference+tolerance, candidate, inside the inclusive interval?).",
      "answer": {
        "kind": "expression",
        "value": "(7, 1/5, 34/5, 36/5, 49/5, No)",
        "explanation": "Check both exact endpoint inequalities; boundary points are within tolerance. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (7, 1/5, 34/5, 36/5, 49/5, No)."
      },
      "skills": [
        "Inclusive Error Interval and a Candidate Boundary"
      ],
      "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": "精確有理數。操作=error-error-interval；資料={\"reference\":[7,1],\"tolerance\":[1,5],\"candidate\":[49,5]}。請依 [reference,tolerance,lower,upper,candidate,in_interval] 欄序回答。",
        "en": "Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. reference=7/1, tolerance=1/5≥0, candidate=49/5. Give (reference, tolerance, reference−tolerance, reference+tolerance, candidate, inside the inclusive interval?)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "reference=7",
      "after": "all values are exact rationals",
      "relation": "implies",
      "rule_id": "exact_error_model",
      "conditions": [],
      "explanation": {
        "zh-Hant": "這是指定參考值下的精確誤差運算，不以浮點四捨五入改變等號邊界。",
        "en": "This is exact error arithmetic against a specified reference,without floating-point rounding that could alter an equality boundary."
      },
      "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": "reference−tolerance := sub(7, 1/5)",
      "after": "reference−tolerance = 34/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先計算閉區間下界。",
        "en": "Compute the closed interval's lower endpoint."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 reference−tolerance，不以小數近似替代。",
          "en": "Recompute reference−tolerance with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "7",
          "1/5"
        ],
        "output": "34/5",
        "label": "reference−tolerance"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "reference+tolerance := sum(7, 1/5)",
      "after": "reference+tolerance = 36/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再計算閉區間上界。",
        "en": "Then compute its upper endpoint."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 reference+tolerance，不以小數近似替代。",
          "en": "Recompute reference+tolerance with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "7",
          "1/5"
        ],
        "output": "36/5",
        "label": "reference+tolerance"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "|candidate−reference|≤tolerance",
      "after": "reference−tolerance≤candidate≤reference+tolerance",
      "relation": "implies",
      "rule_id": "closed_interval_equivalence",
      "conditions": [],
      "explanation": {
        "zh-Hant": "由−epsilon≤誤差≤epsilon，兩邊同加reference得到閉區間；端點相等也通過。",
        "en": "Add the reference to the two-sided signed-error bound to obtain the closed interval;endpoints are included."
      },
      "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": "candidate−reference := sub(49/5, 7)",
      "after": "candidate−reference = 14/5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "另算實際偏差回核區間判斷。",
        "en": "Calculate the actual deviation as a reverse check on interval membership."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 candidate−reference，不以小數近似替代。",
          "en": "Recompute candidate−reference with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "49/5",
          "7"
        ],
        "output": "14/5",
        "label": "candidate−reference"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "candidate=49/5; |deviation|=14/5",
      "after": "inside closed interval: 否",
      "relation": "implies",
      "rule_id": "interval_membership",
      "conditions": [],
      "explanation": {
        "zh-Hant": "誤差絕對值不超過容差與閉區間判斷一致，epsilon=0時只容許reference本身。",
        "en": "The absolute-error and closed-interval tests agree;zero tolerance allows only the reference itself."
      },
      "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",
      "1/5",
      "34/5",
      "36/5",
      "49/5",
      false
    ],
    "signed_error": "14/5",
    "absolute_error": "14/5"
  },
  "conclusion": {
    "canonical_value": "(7, 1/5, 34/5, 36/5, 49/5, 否)",
    "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."
  },
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      "english_rows_path": "public/data/i18n/questions/batch-20261013-advance-20261020-r03-numerical-approximation-error-foundations.en.jsonl"
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        "original_path": "public/data/mko/numerical-approximation-error.json",
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    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
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    "original_candidate_sha256": "696ce3c9ce45652f0e9f3ab7c41a9474b08458f9720448096dc245fb92483aed",
    "original_candidate_line_sha256": "696ce3c9ce45652f0e9f3ab7c41a9474b08458f9720448096dc245fb92483aed"
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  "historical_source_state": {
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      "source_question_state": "candidate",
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      "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."
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  "revision": {
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    "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.",
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    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 6,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
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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."
  }
}