{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261009-advance-20261013-r03-rational-number-0070",
  "pilot_number": "R0011-06420",
  "number": 6420,
  "family_id": "qf-density13-rational-division-guard",
  "title": {
    "zh-Hant": "第 6420 題：有理除法資格與回乘",
    "en": "Question 6420: Rational Division Eligibility and Multiplication Back"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261009-advance-20261013-r03-rational-number-0070",
    "batch_id": "batch-20261009-advance-20261013-r03-rational-number-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-rational-number"
    ],
    "title_zh": "有理除法資格與回乘",
    "stem_zh": "精確有理數協議；q≠0，a=20/-3、b=21/-4。依序答 (b是否非零, a/b或未定義, (a/b)b或未定義)。",
    "answer": {
      "kind": "expression",
      "value": "(是, 80/63, -20/3)",
      "explanation_zh": "除法資格與回乘=(是, 80/63, -20/3)。先檢除數的實值，零除數不回乘補造值；負分母表示有效，輸出分母正且既約，非全部有理數結構證明。"
    },
    "difficulty": {
      "band": "lower_secondary",
      "level": 3,
      "skills": [
        "有理除法資格與回乘"
      ]
    },
    "variant": {
      "family_id": "qf-density13-rational-division-guard",
      "seed": 202610133070,
      "parameters": {
        "a": [
          20,
          -3
        ],
        "b": [
          21,
          -4
        ],
        "operation": "density13_rational_division_guard",
        "domain": "finite_rational_representation_protocol",
        "semantic_template": "qf-density13-rational-division-guard/[\"-20/3\",\"-21/4\"]"
      }
    },
    "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": "2026-10-09",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261009-advance-20261013-r03-rational-number-0070",
    "batch_id": "batch-20261009-advance-20261013-r03-rational-number-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "eb0a517f101fe03ff4ecb9f99af5a9a15dee389a5b159d5e2232f0edf6bf05c1",
      "family_id": "qf-density13-rational-division-guard",
      "seed": 202610133070,
      "semantic_template": "qf-density13-rational-division-guard/[\"-20/3\",\"-21/4\"]"
    },
    "fields": {
      "title": "Rational Division Eligibility and Multiplication Back",
      "stem": "Exact rational representations have nonzero denominators; a=20/-3, b=21/-4. Negative input denominators are allowed. Give (is b nonzero?, a/b or undefined, (a/b)b or undefined). Use exact reduced fractions with positive denominators, retaining /1 for integers.",
      "answer": {
        "kind": "expression",
        "value": "(Yes, 80/63, -20/3)",
        "explanation": "Check the divisor’s value first; a zero divisor does not permit a fabricated quotient or multiplication-back value. The exact result, retaining the requested order, is (Yes, 80/63, -20/3)."
      },
      "skills": [
        "Rational Division Eligibility and Multiplication Back"
      ],
      "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": "精確有理數協議；q≠0，a=20/-3、b=21/-4。依序答 (b是否非零, a/b或未定義, (a/b)b或未定義)。",
        "en": "Exact rational representations have nonzero denominators; a=20/-3, b=21/-4. Negative input denominators are allowed. Give (is b nonzero?, a/b or undefined, (a/b)b or undefined). Use exact reduced fractions with positive denominators, retaining /1 for integers."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "raw a=[20, -3];raw b=[21, -4]",
      "after": "a=-20/3;b=-21/4",
      "relation": "implies",
      "rule_id": "nonzero_signed_rational_representation",
      "conditions": [],
      "explanation": {
        "zh-Hant": "原分母只要求非零，負分母有效；約為正分母既約分數以作精確運算，但保留原始表示與來源。",
        "en": "Original denominators need only be nonzero;negative denominators are valid.Normalize to reduced positive-denominator fractions for exact arithmetic while retaining the original representations and sources."
      },
      "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": "guarded rational quotient := div(-20/3, -21/4)",
      "after": "guarded rational quotient = 80/63",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先已核b非零，才取精確商。",
        "en": "Use the exact quotient only after checking b≠0."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 guarded rational quotient，不以小數近似替代。",
          "en": "Recompute guarded rational quotient with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "-20/3",
          "-21/4"
        ],
        "output": "80/63",
        "label": "guarded rational quotient"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "rational reverse multiplication := mul(80/63, -21/4)",
      "after": "rational reverse multiplication = -20/3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "回乘同一原除數，必須恢復a。",
        "en": "Multiply back by the same divisor to recover a."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 rational reverse multiplication，不以小數近似替代。",
          "en": "Recompute rational reverse multiplication with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "80/63",
          "-21/4"
        ],
        "output": "-20/3",
        "label": "rational reverse multiplication"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "b=-21/4≠0;quotient=80/63",
      "after": "quotient·b=-20/3=a",
      "relation": "implies",
      "rule_id": "rational_nonzero_division_reversal",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只有除數非零時，除法與回乘才是這個核對鏈；不是全部有理數結構的形式證明。",
        "en": "This division/reverse-multiplication chain requires a nonzero divisor;it is not a formal proof of the entire rational-number structure."
      },
      "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": [
      true,
      "80/63",
      "-20/3"
    ]
  },
  "conclusion": {
    "canonical_value": "(是, 80/63, -20/3)",
    "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": {
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      "batch_id": "batch-20261009-advance-20261013-r03-rational-number-foundations",
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      "batch_sha256": "87bdbac39c0ce10a6a71861c70e3b36ec58f0ff44a6888e386a7015f76fc685e",
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      "question_compact_sha256": "eb0a517f101fe03ff4ecb9f99af5a9a15dee389a5b159d5e2232f0edf6bf05c1",
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      "english_manifest_path": "public/data/i18n/editions/batch-20261009-advance-20261013-r03-rational-number-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261009-advance-20261013-r03-rational-number-foundations.en.jsonl"
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        "original_path": "public/data/mko/rational-number.json",
        "english_path": "public/data/i18n/editions/mko-rational-number.en.json"
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    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "5b2ce631281eb24b279908f47c6de66fa60d326520852879d324c4f3ac653970",
    "original_candidate_sha256": "803836b0a218908cd40b758fd5f21376930f090eb0fffdfda468925722804b32",
    "original_candidate_line_sha256": "803836b0a218908cd40b758fd5f21376930f090eb0fffdfda468925722804b32"
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  "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-p12",
    "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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      "question_first_witness_sha256": "93f720df3ab3af13a792a3e6b443728e3c2a56276df839911f36569333236c55"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 4,
    "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."
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}