{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261011-advance-20261015-r01-fraction-0084",
  "pilot_number": "R0012-06984",
  "number": 6984,
  "family_id": "qf-density15-fraction-two-scaled-representations",
  "title": {
    "zh-Hant": "第 6984 題：兩種字面縮放表示",
    "en": "Question 6984: Two Raw Scaled Fraction Representations"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261011-advance-20261015-r01-fraction-0084",
    "batch_id": "batch-20261011-advance-20261015-r01-fraction-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-fraction"
    ],
    "title_zh": "兩種字面縮放表示",
    "stem_zh": "合法分數表示，原分母非零。原n/d中n=-4、d=2，原序縮放因子[s,t]=[2, -1]。求 (原值, 兩個原始[ns,ds]與[nt,dt], 各合法資格, 各規約值或未定義)。使用BigInt保留原始整數產品。",
    "answer": {
      "kind": "expression",
      "value": "(-2/1, [[-8, 4], [4, -2]], [是, 是], [-2/1, -2/1])",
      "explanation_zh": "兩個縮放表示與合法性=(-2/1, [[-8, 4], [4, -2]], [是, 是], [-2/1, -2/1])。同乘非零因子保留值；直接乘出原始兩列，零因子產生0/0。零縮放不給合法分數。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": 2,
      "skills": [
        "兩種字面縮放表示"
      ]
    },
    "variant": {
      "family_id": "qf-density15-fraction-two-scaled-representations",
      "seed": 202610151084,
      "parameters": {
        "n": -4,
        "d": 2,
        "s": 2,
        "t": -1,
        "domain": "finite_fraction_representation_protocol",
        "operation": "density15_fraction_two_scaled_representations",
        "semantic_template": "qf-density15-fraction-two-scaled-representations/[\"-2/1\",[[\"-8\",\"4\"],[\"4\",\"-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": "2026-10-11",
      "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-0084",
    "batch_id": "batch-20261011-advance-20261015-r01-fraction-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "1f40daa08f9abc5bd37b3cf95d4dcbaf162e0cd5d55be0e3771ece6ed220bfeb",
      "family_id": "qf-density15-fraction-two-scaled-representations",
      "seed": 202610151084,
      "semantic_template": "qf-density15-fraction-two-scaled-representations/[\"-2/1\",[[\"-8\",\"4\"],[\"4\",\"-2\"]]]"
    },
    "fields": {
      "title": "Two Raw Scaled Fraction Representations",
      "stem": "Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Original n=-4, d=2; ordered scale factors [s,t]=[2,-1]. Give (original value, raw [ns,ds] and [nt,dt], each representation’s eligibility, reduced values or undefined). Preserve the raw products using exact integer arithmetic.",
      "answer": {
        "kind": "expression",
        "value": "(-2/1, [[-8, 4], [4, -2]], [Yes, Yes], [-2/1, -2/1])",
        "explanation": "A nonzero scale preserves value; a zero scale creates 0/0, which is not a legal representation. Compute raw products before reducing legal representations. The exact result, retaining the requested order, is (-2/1, [[-8, 4], [4, -2]], [Yes, Yes], [-2/1, -2/1])."
      },
      "skills": [
        "Two Raw Scaled Fraction Representations"
      ],
      "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。原n/d中n=-4、d=2，原序縮放因子[s,t]=[2, -1]。求 (原值, 兩個原始[ns,ds]與[nt,dt], 各合法資格, 各規約值或未定義)。使用BigInt保留原始整數產品。",
        "en": "Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Original n=-4, d=2; ordered scale factors [s,t]=[2,-1]. Give (original value, raw [ns,ds] and [nt,dt], each representation’s eligibility, reduced values or undefined). Preserve the raw products using exact integer arithmetic."
      }
    }
  ],
  "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": "original fraction value := div(-4, 2)",
      "after": "original fraction value = -2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "原分母非零，先求原值作非零縮放的回核基準。",
        "en": "With nonzero original denominator,compute the original value as the audit anchor for nonzero scaling."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 original fraction value，不以小數近似替代。",
          "en": "Recompute original fraction value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "-4",
          "2"
        ],
        "output": "-2",
        "label": "original fraction value"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "scaled numerator0 := mul(-4, 2)",
      "after": "scaled numerator0 = -8",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "原分子乘此整數縮放因子，保留完整整數產品。",
        "en": "Multiply the original numerator by this integer scale,retaining the exact integer product."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 scaled numerator0，不以小數近似替代。",
          "en": "Recompute scaled numerator0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-4",
          "2"
        ],
        "output": "-8",
        "label": "scaled numerator0"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "scaled denominator0 := mul(2, 2)",
      "after": "scaled denominator0 = 4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "原分母也乘同一因子，不先約分或刪零。",
        "en": "Multiply the original denominator by the same scale,without prior reduction or deletion of zeros."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 scaled denominator0，不以小數近似替代。",
          "en": "Recompute scaled denominator0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "2"
        ],
        "output": "4",
        "label": "scaled denominator0"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "scaled rational value0 := div(-8, 4)",
      "after": "scaled rational value0 = -2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "新分母非零才可規約為有理值。",
        "en": "Reduce to a rational value only when the new denominator is nonzero."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 scaled rational value0，不以小數近似替代。",
          "en": "Recompute scaled rational value0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "-8",
          "4"
        ],
        "output": "-2",
        "label": "scaled rational value0"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "scale=2;raw scaled pair=[-8, 4]",
      "after": "legal=是;normalized value=-2/1",
      "relation": "implies",
      "rule_id": "scaled_fraction_zero_guard",
      "conditions": [],
      "explanation": {
        "zh-Hant": "非零縮放可約去同一因子並保留值；零縮放產生0/0，分子即使也是0仍不合法，不能約出原值。",
        "en": "A nonzero scale cancels and preserves value;zero scaling produces0/0,which is invalid even with zero numerator and cannot be reduced back to the original value."
      },
      "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": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "scaled numerator1 := mul(-4, -1)",
      "after": "scaled numerator1 = 4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "原分子乘此整數縮放因子，保留完整整數產品。",
        "en": "Multiply the original numerator by this integer scale,retaining the exact integer product."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 scaled numerator1，不以小數近似替代。",
          "en": "Recompute scaled numerator1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-4",
          "-1"
        ],
        "output": "4",
        "label": "scaled numerator1"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "scaled denominator1 := mul(2, -1)",
      "after": "scaled denominator1 = -2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "原分母也乘同一因子，不先約分或刪零。",
        "en": "Multiply the original denominator by the same scale,without prior reduction or deletion of zeros."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 scaled denominator1，不以小數近似替代。",
          "en": "Recompute scaled denominator1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "-1"
        ],
        "output": "-2",
        "label": "scaled denominator1"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "scaled rational value1 := div(4, -2)",
      "after": "scaled rational value1 = -2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "新分母非零才可規約為有理值。",
        "en": "Reduce to a rational value only when the new denominator is nonzero."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 scaled rational value1，不以小數近似替代。",
          "en": "Recompute scaled rational value1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "4",
          "-2"
        ],
        "output": "-2",
        "label": "scaled rational value1"
      }
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "scale=-1;raw scaled pair=[4, -2]",
      "after": "legal=是;normalized value=-2/1",
      "relation": "implies",
      "rule_id": "scaled_fraction_zero_guard",
      "conditions": [],
      "explanation": {
        "zh-Hant": "非零縮放可約去同一因子並保留值；零縮放產生0/0，分子即使也是0仍不合法，不能約出原值。",
        "en": "A nonzero scale cancels and preserves value;zero scaling produces0/0,which is invalid even with zero numerator and cannot be reduced back to the original value."
      },
      "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": [
      "-2",
      [
        [
          -8,
          4
        ],
        [
          4,
          -2
        ]
      ],
      [
        true,
        true
      ],
      [
        "-2",
        "-2"
      ]
    ]
  },
  "conclusion": {
    "canonical_value": "(-2/1, [[-8, 4], [4, -2]], [是, 是], [-2/1, -2/1])",
    "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": {
    "input_commit": "0f7be61e2c22317a4075561dc822c87d881a50db",
    "question_binding": {
      "batch_id": "batch-20261011-advance-20261015-r01-fraction-foundations",
      "batch_path": "public/data/questions/batches/batch-20261011-advance-20261015-r01-fraction-foundations/questions.jsonl",
      "batch_sha256": "33621867facfc78fb3d0ff06bd432e33a58b1514c00596b9334e3a66c8e322fa",
      "original_line": 84,
      "original_line_sha256": "0d898ed53819cdd7b6897292289ee45d79de1c531a64b159f0b5d807214879b3",
      "question_compact_sha256": "1f40daa08f9abc5bd37b3cf95d4dcbaf162e0cd5d55be0e3771ece6ed220bfeb",
      "english_row_compact_sha256": "a8db01632ca993b91e1262ed47c63a0d0e94148d5fe9fa5d901307cefda4eb73",
      "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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        "english_sha256": "ea8b15d28832a77ca243cb2c04a33ac67244aca632c895666d9a33fa9b4982d4",
        "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": "c132db126b58b01d77e01254ea2a9d9c3e51d8a3a15eca75f88d0a656e763c93",
    "original_candidate_line_sha256": "c132db126b58b01d77e01254ea2a9d9c3e51d8a3a15eca75f88d0a656e763c93"
  },
  "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-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": {
      "semantic_review_sha256": "90dbc494b2d34ef51f412dcff63cfcc1a2215d038bff71f38f5dded4c64faf00",
      "question_first_witness_sha256": "b72a3b8bcf10f584396e0dd5198c657b0934a71864e5a3fa6321d7a613aef9f9"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 10,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "3cc680819af02ee9cab6f584642a28d6052e8adc35d6a769e484665ae4af250f",
      "arithmetic_steps": 7
    },
    "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."
  }
}