{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20260930-r01-fraction-equivalence-0011",
  "pilot_number": "R0063-32111",
  "number": 32111,
  "family_id": "qf-fraction-equivalence",
  "title": {
    "zh-Hant": "第 32111 題：分數等值判定",
    "en": "Question 32111: Equivalence of Two Fractions"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20260930-r01-fraction-equivalence-0011",
    "batch_id": "batch-20260930-r01-fraction-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-fraction"
    ],
    "title_zh": "分數等值判定",
    "stem_zh": "整數構成的兩個合法分數 -2/(4) 與 -8/(16) 是否等值？兩個分母均非零。",
    "answer": {
      "kind": "short_text",
      "value": "是",
      "explanation_zh": "合法分母下的兩個交叉乘積為 -32, -32；兩者相等，所以答案為是。"
    },
    "difficulty": {
      "band": "lower_secondary",
      "level": 2,
      "skills": [
        "合法分母",
        "交叉乘積"
      ]
    },
    "variant": {
      "family_id": "qf-fraction-equivalence",
      "seed": 202609301111,
      "parameters": {
        "operation": "equivalence",
        "p": -2,
        "q": 4,
        "r": -8,
        "s": 16,
        "semantic_template": "qf-fraction-equivalence/{\"operation\":\"equivalence\",\"p\":-2,\"q\":4,\"r\":-8,\"s\":16}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "json_schema",
        "producer_local_answer_invariants",
        "explanation_witness_accuracy"
      ],
      "warning_zh": "本題保持 candidate；有限精確計算與產生者自檢不構成人工審查、形式證明或主架構者驗收。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "Codex GPT-6 question producer",
      "generated_at": "2026-09-30",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20260930-r01-fraction-equivalence-0011",
    "batch_id": "batch-20260930-r01-fraction-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "787865eefc0cea16c30ecc013cfa48cc05dcbd5632849f5c7812378db2821f41",
      "family_id": "qf-fraction-equivalence",
      "seed": 202609301111,
      "semantic_template": "qf-fraction-equivalence/{\"operation\":\"equivalence\",\"p\":-2,\"q\":4,\"r\":-8,\"s\":16}"
    },
    "fields": {
      "title": "Equivalence of Two Fractions",
      "stem": "The valid fractions are -2/(4) and -8/(16), with both denominators nonzero. Do they represent the same rational number?",
      "answer": {
        "kind": "short_text",
        "value": "Yes",
        "explanation": "The two cross products are -32 and -32; they are equal, so the answer is Yes."
      },
      "skills": [
        "valid denominators",
        "cross multiplication"
      ],
      "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": "整數構成的兩個合法分數 -2/(4) 與 -8/(16) 是否等值？兩個分母均非零。",
        "en": "The valid fractions are -2/(4) and -8/(16), with both denominators nonzero. Do they represent the same rational number?"
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source roles = [qf-fraction-equivalence, equivalence, no-required-domain, []]",
      "after": "derived = [[exact_equivalence_YesNo_STRING]]",
      "relation": "implies",
      "rule_id": "R63_SOURCE_bank_keys_operation_domain_output_roles_metadata_not_oracle",
      "conditions": [],
      "explanation": {
        "zh-Hant": "BASE來源沒有本輪legacy的通用operation/domain/額外欄位禁令；保metadata而不當答案。PULSE與legacy則核來源精確欄位、operation及domain。不要把tuple、/1或輸入界強套全部題族。",
        "en": "BASE sources do not acquire a generic legacy operation/domain or unknown-metadata ban.Retain metadata WITHOUT using it as an answer oracle.PULSE/legacy enforce their exact source keys,operation/domain.Do not impose universal tuples,/1 or input bounds."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "依原始輸入、型別、論域及明示前提反向代回核對；不是由結論唯一還原前提。來源綁定候選、獨立審查、形式證明與發布各自分開。",
          "en": "Reverse-check ORIGINAL inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "source roles = [-2, 4, [-8, 16]]",
      "after": "derived = [是]",
      "relation": "implies",
      "rule_id": "Fraction_ALL_RAW_SAFE_integer_components_even_zero_den_validity",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先驗全部raw分量的SAFE整數型別，包含q0／0/0資格題的分子。資格題可合法No；其他實際運算才要求所有用到的den非0。",
        "en": "Validate EVERY raw SAFE integer component,including the numerator at q0/0/0 eligibility.A validity task may legally return No;actual arithmetic requires ALL used denominators nonzero."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "依原始輸入、型別、論域及明示前提反向代回核對；不是由結論唯一還原前提。來源綁定候選、獨立審查、形式證明與發布各自分開。",
          "en": "Reverse-check ORIGINAL inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "Fraction ORIGINAL p times s := mul(-2, 16)",
      "after": "Fraction ORIGINAL p times s = -32",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保原操作數與順序作精確整數／有理計算；原輸入界依bank，衍生整數與分數不重套SAFE或外來小界，不用浮點近似。",
        "en": "Use ORIGINAL operands and order for exact integer/rational arithmetic.Raw bounds are bank-specific;derived values do not inherit SAFE or foreign small caps.No floating approximation."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Fraction ORIGINAL p times s，不以小數近似替代。",
          "en": "Recompute Fraction ORIGINAL p times s with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-2",
          "16"
        ],
        "output": "-32",
        "label": "Fraction ORIGINAL p times s"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "Fraction ORIGINAL r times q := mul(-8, 4)",
      "after": "Fraction ORIGINAL r times q = -32",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保原操作數與順序作精確整數／有理計算；原輸入界依bank，衍生整數與分數不重套SAFE或外來小界，不用浮點近似。",
        "en": "Use ORIGINAL operands and order for exact integer/rational arithmetic.Raw bounds are bank-specific;derived values do not inherit SAFE or foreign small caps.No floating approximation."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Fraction ORIGINAL r times q，不以小數近似替代。",
          "en": "Recompute Fraction ORIGINAL r times q with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-8",
          "4"
        ],
        "output": "-32",
        "label": "Fraction ORIGINAL r times q"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "Fraction ORIGINAL cross product difference := sub(-32, -32)",
      "after": "Fraction ORIGINAL cross product difference = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保原操作數與順序作精確整數／有理計算；原輸入界依bank，衍生整數與分數不重套SAFE或外來小界，不用浮點近似。",
        "en": "Use ORIGINAL operands and order for exact integer/rational arithmetic.Raw bounds are bank-specific;derived values do not inherit SAFE or foreign small caps.No floating approximation."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Fraction ORIGINAL cross product difference，不以小數近似替代。",
          "en": "Recompute Fraction ORIGINAL cross product difference with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "-32",
          "-32"
        ],
        "output": "0",
        "label": "Fraction ORIGINAL cross product difference"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "source roles = [-2, 4, -8, 16]",
      "after": "derived = [-32, -32, 0, 是]",
      "relation": "implies",
      "rule_id": "Fraction_EQUIVALENCE_nonzero_dens_EXACT_crossproducts_not_appearance",
      "conditions": [],
      "explanation": {
        "zh-Hant": "兩den非0時p/q=r/s等價於ps=rq，負den也合法。比較完整精確乘積，不用分數外觀或小數近似。",
        "en": "With BOTH denominators nonzero,p/q=r/s IFF ps=rq.Negative denominators are legal.Compare complete exact products,not appearance or decimal approximation."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "依原始輸入、型別、論域及明示前提反向代回核對；不是由結論唯一還原前提。來源綁定候選、獨立審查、形式證明與發布各自分開。",
          "en": "Reverse-check ORIGINAL inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "result": "是"
  },
  "conclusion": {
    "canonical_value": "是",
    "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-20260930-r01-fraction-foundations",
      "batch_path": "public/data/questions/batches/batch-20260930-r01-fraction-foundations/questions.jsonl",
      "batch_sha256": "e2b3f6511abc80a4dc23251454dabd91dee78b87fd84f708bbdd52180842f09f",
      "original_line": 36,
      "original_line_sha256": "a61661617866c0671569c2914556e03c2deadec9cd19e3abd1e67a662f008fa8",
      "question_compact_sha256": "787865eefc0cea16c30ecc013cfa48cc05dcbd5632849f5c7812378db2821f41",
      "english_row_compact_sha256": "f3a0b023590aa7bc566238264e6eb4873fe119a7ca4af8110082e2c56bae1b0e",
      "english_manifest_path": "public/data/i18n/editions/batch-20260930-r01-fraction-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20260930-r01-fraction-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-fraction": {
        "sha256": "aa89902ad405685be331872579647cec33e7ab6792273fe60ad8fefe6129f919",
        "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": "f3271d6047df957dbf3d61120bd3dd374784c828b17849887e14caf59d93f745",
    "original_candidate_sha256": "e29dc45bbf5c8ee5e173002d591f38f044b8cd83110cc79e8eaf1eeed6aa2821",
    "original_candidate_line_sha256": "e29dc45bbf5c8ee5e173002d591f38f044b8cd83110cc79e8eaf1eeed6aa2821"
  },
  "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-20261026-011-p64",
    "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": "acf03b776f4855e3869810bcbecf1116e3a2b11dfb1e2a5dfbcf8135ad24a7d0",
      "question_first_witness_sha256": "eadbbac3220988b52754f32450fb536d627217306d1ce28a49134f56c21bfa88"
    },
    "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": "04d3f7b1b53a2175e0fd8707acf3c0dbdc2b84c660bca89bf8f716e25f5ad050",
      "arithmetic_steps": 3
    },
    "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."
  }
}