{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261011-advance-20261015-r01-fraction-0071",
  "pilot_number": "R0012-06946",
  "number": 6946,
  "family_id": "qf-density15-fraction-inverse-pair-guard",
  "title": {
    "zh-Hant": "第 6946 題：分數乘積與倒數門檻",
    "en": "Question 6946: Fraction Product and Two Reciprocal Guards"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261011-advance-20261015-r01-fraction-0071",
    "batch_id": "batch-20261011-advance-20261015-r01-fraction-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-fraction"
    ],
    "title_zh": "分數乘積與倒數門檻",
    "stem_zh": "合法分數表示，原分母非零。a=(8)/(3)、b=(2)/(-3)。求 (乘積, a與b非零資格, 對應倒數或未定義, 兩倒數乘積或未定義)。",
    "answer": {
      "kind": "expression",
      "value": "(-16/9, [是, 是], [3/8, -3/2], -9/16)",
      "explanation_zh": "分數乘積與倒數資格=(-16/9, [是, 是], [3/8, -3/2], -9/16)。先核兩個數值是否為零，非零才交換規約分子分母求倒數。表示與數值分開，有限分數核對非商集良定性證明。"
    },
    "difficulty": {
      "band": "primary_foundation",
      "level": 3,
      "skills": [
        "分數乘積與倒數門檻"
      ]
    },
    "variant": {
      "family_id": "qf-density15-fraction-inverse-pair-guard",
      "seed": 202610151071,
      "parameters": {
        "a": [
          8,
          3
        ],
        "b": [
          2,
          -3
        ],
        "domain": "finite_fraction_representation_protocol",
        "operation": "density15_fraction_inverse_pair_guard",
        "semantic_template": "qf-density15-fraction-inverse-pair-guard/[\"8/3\",\"-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": "2026-10-11",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261011-advance-20261015-r01-fraction-0071",
    "batch_id": "batch-20261011-advance-20261015-r01-fraction-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "f8d1aecf5dba8a0a6a309a55a9ab443f1780bf087f4855ca298539a11a84483d",
      "family_id": "qf-density15-fraction-inverse-pair-guard",
      "seed": 202610151071,
      "semantic_template": "qf-density15-fraction-inverse-pair-guard/[\"8/3\",\"-2/3\"]"
    },
    "fields": {
      "title": "Fraction Product and Two Reciprocal Guards",
      "stem": "Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. a=8/3, b=2/-3. Give (product, nonzero eligibility list for a and b separately, their reciprocals or undefined, reciprocal product or undefined).",
      "answer": {
        "kind": "expression",
        "value": "(-16/9, [Yes, Yes], [3/8, -3/2], -9/16)",
        "explanation": "Check each operand’s value for zero before taking its reciprocal; exchange normalized numerator/denominator only for a nonzero value. The exact result, retaining the requested order, is (-16/9, [Yes, Yes], [3/8, -3/2], -9/16)."
      },
      "skills": [
        "Fraction Product and Two Reciprocal Guards"
      ],
      "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": "合法分數表示，原分母非零。a=(8)/(3)、b=(2)/(-3)。求 (乘積, a與b非零資格, 對應倒數或未定義, 兩倒數乘積或未定義)。",
        "en": "Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. a=8/3, b=2/-3. Give (product, nonzero eligibility list for a and b separately, their reciprocals or undefined, reciprocal product or undefined)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "original integer numerator/denominator representations;original denominators nonzero",
      "after": "representation and rational value are distinct data",
      "relation": "implies",
      "rule_id": "fraction_representation_value_scope",
      "conditions": [],
      "explanation": {
        "zh-Hant": "負分母表示合法，但約分後的值不取代原始表示；涉及原分子分母的候選算法要用原資料。此有限核對不是商集良定性的一般證明。",
        "en": "Negative denominators are valid,but reduced values do not replace original representations.Candidate algorithms about raw numerators/denominators must use those raw data.This finite audit is not a general proof of quotient well-definedness."
      },
      "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": "fraction product := mul(8/3, -2/3)",
      "after": "fraction product = -16/9",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先乘兩個合法分數的實值，零值仍可相乘。",
        "en": "Multiply the values of the two valid fractions;zero values may still be multiplied."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 fraction product，不以小數近似替代。",
          "en": "Recompute fraction product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "8/3",
          "-2/3"
        ],
        "output": "-16/9",
        "label": "fraction product"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "fraction reciprocal0 := div(1, 8/3)",
      "after": "fraction reciprocal0 = 3/8",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只有分數的實值非零才有乘法倒數。",
        "en": "A multiplicative reciprocal exists only when the fraction's value is nonzero."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 fraction reciprocal0，不以小數近似替代。",
          "en": "Recompute fraction reciprocal0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "1",
          "8/3"
        ],
        "output": "3/8",
        "label": "fraction reciprocal0"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "reciprocal reverse product0 := mul(8/3, 3/8)",
      "after": "reciprocal reverse product0 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "原值乘倒數反核，必須等於1。",
        "en": "Multiply the value by its reciprocal to check that the result is1."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 reciprocal reverse product0，不以小數近似替代。",
          "en": "Recompute reciprocal reverse product0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "8/3",
          "3/8"
        ],
        "output": "1",
        "label": "reciprocal reverse product0"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "value=8/3;nonzero=是",
      "after": "reciprocal=3/8",
      "relation": "implies",
      "rule_id": "fraction_reciprocal_value_guard",
      "conditions": [],
      "explanation": {
        "zh-Hant": "判斷的是值是否為0，不是原分母符號；0/負分母仍是合法零值，但沒有倒數。",
        "en": "Test whether the value is0,not the sign of the original denominator.A zero numerator over a negative nonzero denominator is valid zero,but has no reciprocal."
      },
      "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": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "fraction reciprocal1 := div(1, -2/3)",
      "after": "fraction reciprocal1 = -3/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只有分數的實值非零才有乘法倒數。",
        "en": "A multiplicative reciprocal exists only when the fraction's value is nonzero."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 fraction reciprocal1，不以小數近似替代。",
          "en": "Recompute fraction reciprocal1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "1",
          "-2/3"
        ],
        "output": "-3/2",
        "label": "fraction reciprocal1"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "reciprocal reverse product1 := mul(-2/3, -3/2)",
      "after": "reciprocal reverse product1 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "原值乘倒數反核，必須等於1。",
        "en": "Multiply the value by its reciprocal to check that the result is1."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 reciprocal reverse product1，不以小數近似替代。",
          "en": "Recompute reciprocal reverse product1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-2/3",
          "-3/2"
        ],
        "output": "1",
        "label": "reciprocal reverse product1"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "value=-2/3;nonzero=是",
      "after": "reciprocal=-3/2",
      "relation": "implies",
      "rule_id": "fraction_reciprocal_value_guard",
      "conditions": [],
      "explanation": {
        "zh-Hant": "判斷的是值是否為0，不是原分母符號；0/負分母仍是合法零值，但沒有倒數。",
        "en": "Test whether the value is0,not the sign of the original denominator.A zero numerator over a negative nonzero denominator is valid zero,but has no reciprocal."
      },
      "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": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "product of two reciprocals := mul(3/8, -3/2)",
      "after": "product of two reciprocals = -9/16",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "兩倒數都已定義時才相乘。",
        "en": "Multiply the reciprocals only when both are defined."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 product of two reciprocals，不以小數近似替代。",
          "en": "Recompute product of two reciprocals with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "3/8",
          "-3/2"
        ],
        "output": "-9/16",
        "label": "product of two reciprocals"
      }
    }
  ],
  "math_model": {
    "result": [
      "-16/9",
      [
        true,
        true
      ],
      [
        "3/8",
        "-3/2"
      ],
      "-9/16"
    ]
  },
  "conclusion": {
    "canonical_value": "(-16/9, [是, 是], [3/8, -3/2], -9/16)",
    "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": {
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      "batch_path": "public/data/questions/batches/batch-20261011-advance-20261015-r01-fraction-foundations/questions.jsonl",
      "batch_sha256": "33621867facfc78fb3d0ff06bd432e33a58b1514c00596b9334e3a66c8e322fa",
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      "question_compact_sha256": "f8d1aecf5dba8a0a6a309a55a9ab443f1780bf087f4855ca298539a11a84483d",
      "english_row_compact_sha256": "28367da178c866413717dade4414e367fe4fc805a1a712bc301881005103ec6d",
      "english_manifest_path": "public/data/i18n/editions/batch-20261011-advance-20261015-r01-fraction-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261011-advance-20261015-r01-fraction-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-fraction": {
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        "original_path": "public/data/mko/fraction.json",
        "english_path": "public/data/i18n/editions/mko-fraction.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "c5c612ca91797c328f4489939ec6f7d885385e63ea21b334c09c421f17933d99",
    "original_candidate_sha256": "f9dc75678845a018f3f202e75c2a6a67d0d41a02cda1c7576cb9dde3caed4f88",
    "original_candidate_line_sha256": "f9dc75678845a018f3f202e75c2a6a67d0d41a02cda1c7576cb9dde3caed4f88"
  },
  "historical_source_state": {
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      "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": {
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      "question_first_witness_sha256": "b72a3b8bcf10f584396e0dd5198c657b0934a71864e5a3fa6321d7a613aef9f9"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 9,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
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      "arithmetic_steps": 6
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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."
  }
}