{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261008-advance-20261011-r02-arithmetic-operations-0034",
  "pilot_number": "R0008-04659",
  "number": 4659,
  "family_id": "qf-density11-arithmetic-parentheses-pair",
  "title": {
    "zh-Hant": "第 4659 題：括號位置的雙式值",
    "en": "Question 4659: Two Explicit Parenthesis Orders"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261008-advance-20261011-r02-arithmetic-operations-0034",
    "batch_id": "batch-20261008-advance-20261011-r02-arithmetic-operations-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-arithmetic-operations"
    ],
    "title_zh": "括號位置的雙式值",
    "stem_zh": "有理數Q；a=-2/1、b=-7/3、c=9/2。依序求 ((a+b)c, a+bc)，保留括號。",
    "answer": {
      "kind": "expression",
      "value": "(-39/2, -25/2)",
      "explanation_zh": "括號雙式值=(-39/2, -25/2)。第一式先加後乘，第二式先乘後加；零乘數與一乘數可使某些項消失，不因此改計算次序。"
    },
    "difficulty": {
      "band": "lower_secondary",
      "level": 2,
      "skills": [
        "括號位置的雙式值"
      ]
    },
    "variant": {
      "family_id": "qf-density11-arithmetic-parentheses-pair",
      "seed": 202610112034,
      "parameters": {
        "a": [
          -4,
          2
        ],
        "b": [
          -7,
          3
        ],
        "c": [
          9,
          2
        ],
        "operation": "density11_arithmetic_parentheses_pair",
        "domain": "rational_arithmetic_slice",
        "semantic_template": "qf-density11-arithmetic-parentheses-pair/[\"-2/1\",\"-7/3\",\"9/2\"]"
      }
    },
    "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-0034",
    "batch_id": "batch-20261008-advance-20261011-r02-arithmetic-operations-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "0a6298e6e4c345a236f1ea5b5977a7bc4c8fc9ce83770b632f327c9442deac48",
      "family_id": "qf-density11-arithmetic-parentheses-pair",
      "seed": 202610112034,
      "semantic_template": "qf-density11-arithmetic-parentheses-pair/[\"-2/1\",\"-7/3\",\"9/2\"]"
    },
    "fields": {
      "title": "Two Explicit Parenthesis Orders",
      "stem": "All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=-4/2, b=-7/3, c=9/2. Give ((a+b)c, a+bc), preserving parentheses. Use exact reduced fractions with positive denominators, retaining /1 for integers.",
      "answer": {
        "kind": "expression",
        "value": "(-39/2, -25/2)",
        "explanation": "The first expression adds before multiplying, while the second multiplies before adding; zero or unit factors do not change the declared evaluation order. The exact result, retaining the requested order, is (-39/2, -25/2)."
      },
      "skills": [
        "Two Explicit Parenthesis Orders"
      ],
      "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=-2/1、b=-7/3、c=9/2。依序求 ((a+b)c, a+bc)，保留括號。",
        "en": "All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=-4/2, b=-7/3, c=9/2. Give ((a+b)c, a+bc), preserving parentheses. 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 inside parentheses := sum(-2, -7/3)",
      "after": "a+b inside parentheses = -13/3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第一式先完成括號內加法。",
        "en": "For the first expression,evaluate the parenthesized sum first."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 a+b inside parentheses，不以小數近似替代。",
          "en": "Recompute a+b inside parentheses with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "-7/3"
        ],
        "output": "-13/3",
        "label": "a+b inside parentheses"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "(a+b)c := mul(-13/3, 9/2)",
      "after": "(a+b)c = -39/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "把完整括號值乘c。",
        "en": "Multiply the entire parenthesized value by c."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 (a+b)c，不以小數近似替代。",
          "en": "Recompute (a+b)c with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-13/3",
          "9/2"
        ],
        "output": "-39/2",
        "label": "(a+b)c"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "bc before addition := mul(-7/3, 9/2)",
      "after": "bc before addition = -21/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第二式先做乘法bc。",
        "en": "For the second expression,compute bc first."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 bc before addition，不以小數近似替代。",
          "en": "Recompute bc before addition with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-7/3",
          "9/2"
        ],
        "output": "-21/2",
        "label": "bc before addition"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "a+bc := sum(-2, -21/2)",
      "after": "a+bc = -25/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再加a；不能沿用第一式的分組。",
        "en": "Then add a without borrowing the first expression's grouping."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 a+bc，不以小數近似替代。",
          "en": "Recompute a+bc with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2",
          "-21/2"
        ],
        "output": "-25/2",
        "label": "a+bc"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "((a+b)c,a+bc)=(-39/2,-25/2)",
      "after": "coinciding special values do not change the expression trees",
      "relation": "implies",
      "rule_id": "parentheses_preserve_evaluation_tree",
      "conditions": [],
      "explanation": {
        "zh-Hant": "特殊零或一可能使兩式數值相同，仍不構成任意移除括號的許可。",
        "en": "Special zeros or ones may make the values coincide,but do not permit arbitrary removal of parentheses."
      },
      "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": [
      "-39/2",
      "-25/2"
    ]
  },
  "conclusion": {
    "canonical_value": "(-39/2, -25/2)",
    "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": {
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      "original_line": 34,
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      "question_compact_sha256": "0a6298e6e4c345a236f1ea5b5977a7bc4c8fc9ce83770b632f327c9442deac48",
      "english_row_compact_sha256": "ecd945748fb5b9d5891aa426253b64e5737135cc91727d89725329f65e970395",
      "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"
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    "mko_bindings": {
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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": "5247a357770f9c41961bb18d15e801c1b071279e1a1d160bfd4dd6160a8d69ab",
    "original_candidate_line_sha256": "5247a357770f9c41961bb18d15e801c1b071279e1a1d160bfd4dd6160a8d69ab"
  },
  "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": "90d8d97349017fc78a78f4deef31ac247d86e37c5cd79406d36f227a1e2b5b7d",
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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."
  }
}