{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261013-advance-20261021-r02-diagnostic-mathematical-assessment-0047",
  "pilot_number": "R0003-02122",
  "number": 2122,
  "family_id": "qf-app21-diagnostic-fraction-equivalence-record",
  "title": {
    "zh-Hant": "第 2122 題：合成診斷記錄",
    "en": "Question 2122: Synthetic Fraction-Equivalence Response with Nonzero Denominators"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261013-advance-20261021-r02-diagnostic-mathematical-assessment-0047",
    "batch_id": "batch-20261013-advance-20261021-r02-diagnostic-mathematical-assessment-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-diagnostic-mathematical-assessment"
    ],
    "title_zh": "合成診斷記錄",
    "stem_zh": "合成作答紀錄與明示規則。操作=diagnostic-fraction-equivalence-record；資料={\"p\":13,\"q\":2,\"r\":13,\"s\":1,\"response\":false}。請依 [p,q,r,s,left_cross,right_cross,equivalent,response,match,rule_code] 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(13, 2, 13, 1, 13, 26, 否, 否, 是, matches_rule_expected)",
      "explanation_zh": "應用擴展核對=(13, 2, 13, 1, 13, 26, 否, 否, 是, matches_rule_expected)；依完整明示資料重算；完整明示的有限／精確切片不是一般定理或人員判定；只記錄規則符合狀態；不診斷真實學生或學習效果。"
    },
    "difficulty": {
      "band": "upper_secondary",
      "level": 3,
      "skills": [
        "合成診斷記錄"
      ]
    },
    "variant": {
      "family_id": "qf-app21-diagnostic-fraction-equivalence-record",
      "seed": 202610212047,
      "parameters": {
        "p": 13,
        "q": 2,
        "r": 13,
        "s": 1,
        "response": false,
        "domain": "Synthetic_rule_diagnostic_record",
        "operation": "diagnostic-fraction-equivalence-record",
        "semantic_template": "qf-app21-diagnostic-fraction-equivalence-record/{\"p\":13,\"q\":2,\"r\":13,\"response\":false,\"s\":1}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_finite_model",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、真實學生診斷、經驗預測、Lean執行或形式Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-13",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261013-advance-20261021-r02-diagnostic-mathematical-assessment-0047",
    "batch_id": "batch-20261013-advance-20261021-r02-diagnostic-mathematical-assessment-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "96ea478d37f0441541aa0512b8772ab21b9263ba78ed3cbe14274e146380750f",
      "family_id": "qf-app21-diagnostic-fraction-equivalence-record",
      "seed": 202610212047,
      "semantic_template": "qf-app21-diagnostic-fraction-equivalence-record/{\"p\":13,\"q\":2,\"r\":13,\"response\":false,\"s\":1}"
    },
    "fields": {
      "title": "Synthetic Fraction-Equivalence Response with Nonzero Denominators",
      "stem": "Use a synthetic response record and the explicitly declared mathematical rule. A match code reports only rule agreement, not the ability, personality, learning difficulty, cause of error or learning outcome of a real student. Fractions p/q and r/s have p=13, q=2≠0, r=13, s=1≠0; Boolean response=false. Give (p,q,r,s,p*s,r*q,equivalent?,response,match?,rule code). Preserve the signed original numerator/denominator values in these fields.",
      "answer": {
        "kind": "expression",
        "value": "(13, 2, 13, 1, 13, 26, No, No, Yes, matches_rule_expected)",
        "explanation": "Cross products test equivalence exactly, including negative denominators; compare that truth value with the synthetic response without replacing the record’s raw fractions. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (13, 2, 13, 1, 13, 26, No, No, Yes, matches_rule_expected)."
      },
      "skills": [
        "Synthetic Fraction-Equivalence Response with Nonzero Denominators"
      ],
      "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": "合成作答紀錄與明示規則。操作=diagnostic-fraction-equivalence-record；資料={\"p\":13,\"q\":2,\"r\":13,\"s\":1,\"response\":false}。請依 [p,q,r,s,left_cross,right_cross,equivalent,response,match,rule_code] 欄序回答。",
        "en": "Use a synthetic response record and the explicitly declared mathematical rule. A match code reports only rule agreement, not the ability, personality, learning difficulty, cause of error or learning outcome of a real student. Fractions p/q and r/s have p=13, q=2≠0, r=13, s=1≠0; Boolean response=false. Give (p,q,r,s,p*s,r*q,equivalent?,response,match?,rule code). Preserve the signed original numerator/denominator values in these fields."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "synthetic response record and a declared mathematical rule",
      "after": "check rule agreement,not a learner diagnosis",
      "relation": "implies",
      "rule_id": "synthetic_record_scope",
      "conditions": [],
      "explanation": {
        "zh-Hant": "這些是合成紀錄；不從單一作答推論真實學生能力、心理或學習效果。",
        "en": "These are synthetic records;one response does not diagnose a real learner's ability,psychology or learning outcome."
      },
      "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": "denominators=2,1, both nonzero",
      "after": "a/b=c/d iff a·d=c·b",
      "relation": "implies",
      "rule_id": "fraction_equality_guard",
      "conditions": [],
      "explanation": {
        "zh-Hant": "等式可乘同一非零分母乘積，即使分母為負也不需要改變等號。",
        "en": "Multiply an equality by the same nonzero denominator product;negative denominators do not reverse equality."
      },
      "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": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "left cross-product := mul(13, 1)",
      "after": "left cross-product = 13",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "左分子乘右分母，保留原始整數。",
        "en": "Multiply the left numerator by the right denominator,retaining the original integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left cross-product，不以小數近似替代。",
          "en": "Recompute left cross-product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "13",
          "1"
        ],
        "output": "13",
        "label": "left cross-product"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "right cross-product := mul(13, 2)",
      "after": "right cross-product = 26",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "右分子乘左分母。",
        "en": "Multiply the right numerator by the left denominator."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 right cross-product，不以小數近似替代。",
          "en": "Recompute right cross-product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "13",
          "2"
        ],
        "output": "26",
        "label": "right cross-product"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "cross-products=13,26; equivalent=否",
      "after": "response=否; match=是",
      "relation": "implies",
      "rule_id": "boolean_response_match",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先求正確等價真假，再看作答真假是否符合；False是合法作答，不是缺失。",
        "en": "Determine the correct equivalence truth first,then compare the Boolean response;False is a valid response,not missing data."
      },
      "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": "rule agreement=是",
      "after": "record code=matches_rule_expected",
      "relation": "implies",
      "rule_id": "diagnostic_rule_code",
      "conditions": [],
      "explanation": {
        "zh-Hant": "代碼只表示本紀錄與明示規則是否符合，不冒充人員評分或一般學習診斷。",
        "en": "The code records agreement with the stated rule,not a person-level score or general learning diagnosis."
      },
      "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": [
      13,
      2,
      13,
      1,
      "13",
      "26",
      false,
      false,
      true,
      "matches_rule_expected"
    ]
  },
  "conclusion": {
    "canonical_value": "(13, 2, 13, 1, 13, 26, 否, 否, 是, matches_rule_expected)",
    "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_path": "public/data/questions/batches/batch-20261013-advance-20261021-r02-diagnostic-mathematical-assessment-foundations/questions.jsonl",
      "batch_sha256": "4e15298c1f1f0564f4a1482fd93ca7d0b8038c5e69a4cf31de050461f82504f2",
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      "original_line_sha256": "f3b6d5e7aff948d54eeca4950e7855872d3dd6c8349cd8988741e19ee469f56a",
      "question_compact_sha256": "96ea478d37f0441541aa0512b8772ab21b9263ba78ed3cbe14274e146380750f",
      "english_row_compact_sha256": "c1fc23fa10667ae4c60c3b54fc3e6a711a7ab0c32254c4aad53b4d9c27e52444",
      "english_manifest_path": "public/data/i18n/editions/batch-20261013-advance-20261021-r02-diagnostic-mathematical-assessment-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261013-advance-20261021-r02-diagnostic-mathematical-assessment-foundations.en.jsonl"
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    "mko_bindings": {
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        "english_sha256": "0cc9c352db42a18773ad0b54fa57bb86d3b959d8a0ae8ba17137e325e9b357b3",
        "original_path": "public/data/mko/diagnostic-mathematical-assessment.json",
        "english_path": "public/data/i18n/editions/mko-diagnostic-mathematical-assessment.en.json"
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    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "09b39eb68bdeb3e1e4277f359de31284852356add54b99a43cf1e38a33d69d15",
    "original_candidate_sha256": "279546c722635dc26494b93166cec620c1fa6fed5e2defdb52e3b7f288ec3a11",
    "original_candidate_line_sha256": "279546c722635dc26494b93166cec620c1fa6fed5e2defdb52e3b7f288ec3a11"
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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-20261026-004-p4",
    "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": "2b93d7d38874472443c7141d10c1e292c6b086615cac7152f7cacf06633b0aba",
      "arithmetic_steps": 2
    },
    "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."
  }
}