{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261008-advance-20261011-r02-arithmetic-operations-0021",
  "pilot_number": "R0008-04646",
  "number": 4646,
  "family_id": "qf-density11-arithmetic-ordered-four-results",
  "title": {
    "zh-Hant": "第 4646 題：有序四則運算結果",
    "en": "Question 4646: Ordered Four Rational Operations"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261008-advance-20261011-r02-arithmetic-operations-0021",
    "batch_id": "batch-20261008-advance-20261011-r02-arithmetic-operations-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-arithmetic-operations"
    ],
    "title_zh": "有序四則運算結果",
    "stem_zh": "有理數Q；a=8/3、b=21/2。依序求 (a+b, a−b, ab, a/b)，未定義的欄位寫未定義。",
    "answer": {
      "kind": "expression",
      "value": "(79/6, -47/6, 28/1, 16/63)",
      "explanation_zh": "有序四則結果=(79/6, -47/6, 28/1, 16/63)。以正分母分數精確加減乘除，除零未定義；結果不使用自然數截斷運算。"
    },
    "difficulty": {
      "band": "lower_secondary",
      "level": 3,
      "skills": [
        "有序四則運算結果"
      ]
    },
    "variant": {
      "family_id": "qf-density11-arithmetic-ordered-four-results",
      "seed": 202610112021,
      "parameters": {
        "a": [
          8,
          3
        ],
        "b": [
          21,
          2
        ],
        "operation": "density11_arithmetic_ordered_four_results",
        "domain": "rational_arithmetic_slice",
        "semantic_template": "qf-density11-arithmetic-ordered-four-results/[\"8/3\",\"21/2\"]"
      }
    },
    "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": "2026-10-08",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261008-advance-20261011-r02-arithmetic-operations-0021",
    "batch_id": "batch-20261008-advance-20261011-r02-arithmetic-operations-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "22071df6d5b0f84e98f66599ef7895a84cb93060755204f148da991040ccceed",
      "family_id": "qf-density11-arithmetic-ordered-four-results",
      "seed": 202610112021,
      "semantic_template": "qf-density11-arithmetic-ordered-four-results/[\"8/3\",\"21/2\"]"
    },
    "fields": {
      "title": "Ordered Four Rational Operations",
      "stem": "All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=8/3, b=21/2. Give (a+b, a−b, ab, a/b), using undefined for an undefined field. Use exact reduced fractions with positive denominators, retaining /1 for integers.",
      "answer": {
        "kind": "expression",
        "value": "(79/6, -47/6, 28/1, 16/63)",
        "explanation": "Perform exact addition, subtraction, multiplication, and guarded division in that order; division by zero is undefined. The exact result, retaining the requested order, is (79/6, -47/6, 28/1, 16/63)."
      },
      "skills": [
        "Ordered Four Rational Operations"
      ],
      "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": "有理數Q；a=8/3、b=21/2。依序求 (a+b, a−b, ab, a/b)，未定義的欄位寫未定義。",
        "en": "All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=8/3, b=21/2. Give (a+b, a−b, ab, a/b), using undefined for an undefined field. Use exact reduced fractions with positive denominators, retaining /1 for integers."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "exact arithmetic in Q;N={0,1,2,...} when membership is asked",
      "after": "do not truncate subtraction or rational division",
      "relation": "implies",
      "rule_id": "exact_rational_arithmetic_scope",
      "conditions": [],
      "explanation": {
        "zh-Hant": "四則結果先在Q中精確計算，再判N資格；負差不截斷為0，非整數商不做整除，除零仍未定義。",
        "en": "Compute exact results in Q before testing N-membership;do not clamp negative differences to0 or replace rational division by integer division.Division by0 remains undefined."
      },
      "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": "a+b := sum(8/3, 21/2)",
      "after": "a+b = 79/6",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "加法按精確分數計算。",
        "en": "Compute addition as exact fractions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 a+b，不以小數近似替代。",
          "en": "Recompute a+b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "8/3",
          "21/2"
        ],
        "output": "79/6",
        "label": "a+b"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "a−b := sub(8/3, 21/2)",
      "after": "a−b = -47/6",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "減法保持原次序。",
        "en": "Retain operand order in subtraction."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 a−b，不以小數近似替代。",
          "en": "Recompute a−b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "8/3",
          "21/2"
        ],
        "output": "-47/6",
        "label": "a−b"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "ab := mul(8/3, 21/2)",
      "after": "ab = 28",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "乘法仍在Q中，不取近似值。",
        "en": "Multiply in Q without approximation."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 ab，不以小數近似替代。",
          "en": "Recompute ab with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "8/3",
          "21/2"
        ],
        "output": "28",
        "label": "ab"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "a/b := div(8/3, 21/2)",
      "after": "a/b = 16/63",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "b非零才乘其倒數。",
        "en": "Only a nonzero b permits division by its reciprocal."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 a/b，不以小數近似替代。",
          "en": "Recompute a/b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "8/3",
          "21/2"
        ],
        "output": "16/63",
        "label": "a/b"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "b=21/2",
      "after": "division defined=是;quotient=16/63",
      "relation": "implies",
      "rule_id": "division_zero_guard",
      "conditions": [],
      "explanation": {
        "zh-Hant": "除數0不是一個很小的非零分母，也不能把0/0或a/0指定成0。",
        "en": "A zero divisor is not a small nonzero denominator;neither0/0 nor a/0 may be assigned0."
      },
      "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": [
      "79/6",
      "-47/6",
      "28",
      "16/63"
    ],
    "division_defined": true
  },
  "conclusion": {
    "canonical_value": "(79/6, -47/6, 28/1, 16/63)",
    "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-20261008-advance-20261011-r02-arithmetic-operations-foundations",
      "batch_path": "public/data/questions/batches/batch-20261008-advance-20261011-r02-arithmetic-operations-foundations/questions.jsonl",
      "batch_sha256": "f41f31790791f96d2b83340fd50e7de66aced987af07332047451f40a2281fde",
      "original_line": 21,
      "original_line_sha256": "02b18b42d13f398366f3789b9fe71a91cac4d5d06e8f87c0a5a6451fc526afee",
      "question_compact_sha256": "22071df6d5b0f84e98f66599ef7895a84cb93060755204f148da991040ccceed",
      "english_row_compact_sha256": "8f5c1047b1d44067b34c0aa13d33b5eaf5caa7add543f57868206013cb35190c",
      "english_manifest_path": "public/data/i18n/editions/batch-20261008-advance-20261011-r02-arithmetic-operations-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261008-advance-20261011-r02-arithmetic-operations-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-arithmetic-operations": {
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        "original_path": "public/data/mko/arithmetic-operations.json",
        "english_path": "public/data/i18n/editions/mko-arithmetic-operations.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "6dc34e24b0f43c50e49365ca1a1153175f8661d5290c9236df5527bb047157eb",
    "original_candidate_sha256": "6fa2ead7197229a30dd9e081df9ccf65eb2c6ddcf9126d35dca5fbb6b8adca17",
    "original_candidate_line_sha256": "6fa2ead7197229a30dd9e081df9ccf65eb2c6ddcf9126d35dca5fbb6b8adca17"
  },
  "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-20261026-004-p9",
    "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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    },
    "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": "60dc523d67143410a2d77c355fd2e92e45761ba0805c23d35f1f974e6aa90370",
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    },
    "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."
  }
}