{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261008-advance-20261011-r02-arithmetic-operations-0069",
  "pilot_number": "R0008-04619",
  "number": 4619,
  "family_id": "qf-density11-arithmetic-inverse-candidate",
  "title": {
    "zh-Hant": "第 4619 題：乘法逆元資格與乘積缺陷",
    "en": "Question 4619: Inverse Candidate and Multiplicative Defect"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261008-advance-20261011-r02-arithmetic-operations-0069",
    "batch_id": "batch-20261008-advance-20261011-r02-arithmetic-operations-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-arithmetic-operations"
    ],
    "title_zh": "乘法逆元資格與乘積缺陷",
    "stem_zh": "有理數Q；a=19/2、候選r=2/19。依序答 (r是否為a的乘法逆元, ar−1)，第二欄以精確分數表示。",
    "answer": {
      "kind": "expression",
      "value": "(是, 0/1)",
      "explanation_zh": "逆元資格與乘積缺陷=(是, 0/1)。零沒有乘法逆元；非零a也須乘積恰為一才通過，缺陷保留正負號。"
    },
    "difficulty": {
      "band": "lower_secondary",
      "level": 3,
      "skills": [
        "乘法逆元資格與乘積缺陷"
      ]
    },
    "variant": {
      "family_id": "qf-density11-arithmetic-inverse-candidate",
      "seed": 202610112069,
      "parameters": {
        "a": [
          19,
          2
        ],
        "r": [
          2,
          19
        ],
        "operation": "density11_arithmetic_inverse_candidate",
        "domain": "rational_arithmetic_slice",
        "semantic_template": "qf-density11-arithmetic-inverse-candidate/[\"19/2\",\"2/19\"]"
      }
    },
    "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": "2026-10-08",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261008-advance-20261011-r02-arithmetic-operations-0069",
    "batch_id": "batch-20261008-advance-20261011-r02-arithmetic-operations-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "4320b53cdeef61c032efa9580784c0922ca1ec7d483203b45dc0d509abbef4fe",
      "family_id": "qf-density11-arithmetic-inverse-candidate",
      "seed": 202610112069,
      "semantic_template": "qf-density11-arithmetic-inverse-candidate/[\"19/2\",\"2/19\"]"
    },
    "fields": {
      "title": "Inverse Candidate and Multiplicative Defect",
      "stem": "All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=19/2, candidate r=2/19. Give (is r a multiplicative inverse of a?, ar−1). Use exact reduced fractions with positive denominators, retaining /1 for integers.",
      "answer": {
        "kind": "expression",
        "value": "(Yes, 0/1)",
        "explanation": "Zero has no multiplicative inverse; a nonzero operand still requires the product to equal exactly one. Keep the defect’s sign. The exact result, retaining the requested order, is (Yes, 0/1)."
      },
      "skills": [
        "Inverse Candidate and Multiplicative Defect"
      ],
      "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": "有理數Q；a=19/2、候選r=2/19。依序答 (r是否為a的乘法逆元, ar−1)，第二欄以精確分數表示。",
        "en": "All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=19/2, candidate r=2/19. Give (is r a multiplicative inverse of a?, ar−1). Use exact reduced fractions with positive denominators, retaining /1 for integers."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "exact arithmetic in Q;N={0,1,2,...} when membership is asked",
      "after": "do not truncate subtraction or rational division",
      "relation": "implies",
      "rule_id": "exact_rational_arithmetic_scope",
      "conditions": [],
      "explanation": {
        "zh-Hant": "四則結果先在Q中精確計算，再判N資格；負差不截斷為0，非整數商不做整除，除零仍未定義。",
        "en": "Compute exact results in Q before testing N-membership;do not clamp negative differences to0 or replace rational division by integer division.Division by0 remains undefined."
      },
      "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": "candidate inverse product := mul(19/2, 2/19)",
      "after": "candidate inverse product = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "乘法逆元必須與原數相乘得到1。",
        "en": "A multiplicative inverse must multiply the original number to1."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 candidate inverse product，不以小數近似替代。",
          "en": "Recompute candidate inverse product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "19/2",
          "2/19"
        ],
        "output": "1",
        "label": "candidate inverse product"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "inverse signed defect := sub(1, 1)",
      "after": "inverse signed defect = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按ar−1方向計算有號缺陷。",
        "en": "Compute the signed defect ar−1 in that order."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 inverse signed defect，不以小數近似替代。",
          "en": "Recompute inverse signed defect with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "1",
          "1"
        ],
        "output": "0",
        "label": "inverse signed defect"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "a=19/2;r=2/19;ar=1",
      "after": "inverse valid=是;ar−1=0",
      "relation": "implies",
      "rule_id": "multiplicative_inverse_zero_boundary",
      "conditions": [],
      "explanation": {
        "zh-Hant": "a=0時任何候選乘積均0，不可能成逆元；a非零也仍須實際檢查乘積1。",
        "en": "For a=0 every candidate product is0,so no inverse exists;even with nonzero a,the candidate must actually produce1."
      },
      "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,
      "0"
    ]
  },
  "conclusion": {
    "canonical_value": "(是, 0/1)",
    "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-20261008-advance-20261011-r02-arithmetic-operations-foundations",
      "batch_path": "public/data/questions/batches/batch-20261008-advance-20261011-r02-arithmetic-operations-foundations/questions.jsonl",
      "batch_sha256": "f41f31790791f96d2b83340fd50e7de66aced987af07332047451f40a2281fde",
      "original_line": 69,
      "original_line_sha256": "39e50f9d9c7ff819b6ebe5102db7e2eb163f302a35d717ca5dc92f69032d116d",
      "question_compact_sha256": "4320b53cdeef61c032efa9580784c0922ca1ec7d483203b45dc0d509abbef4fe",
      "english_row_compact_sha256": "f969e288dd424450ea07c5b5de3523773d71a04ee9653e55f4993dd76dd49dc0",
      "english_manifest_path": "public/data/i18n/editions/batch-20261008-advance-20261011-r02-arithmetic-operations-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261008-advance-20261011-r02-arithmetic-operations-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-arithmetic-operations": {
        "sha256": "452d77028e2659b860c94200c1de159c4184dde86d6d73950e1998a0080e265a",
        "english_sha256": "cd55f6d985015e73e560232ec9e4f77f3559c06dc72f0cf006f7083206a54169",
        "original_path": "public/data/mko/arithmetic-operations.json",
        "english_path": "public/data/i18n/editions/mko-arithmetic-operations.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "6dc34e24b0f43c50e49365ca1a1153175f8661d5290c9236df5527bb047157eb",
    "original_candidate_sha256": "6c9dece69d8828e0b10b656d71c1e0293ca6e2321cfe0eb5094048508c8e85f5",
    "original_candidate_line_sha256": "6c9dece69d8828e0b10b656d71c1e0293ca6e2321cfe0eb5094048508c8e85f5"
  },
  "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-p9",
    "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": "b6e05a3e8193a93db505dbf337528fca72e2e41528b1e92b698a3f0683d74289",
      "question_first_witness_sha256": "caa8693baf0b91acd09a20215d1aa6954ae513fc0b0814f26ac1c8c834b60d67"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 4,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "09c6aa2530dbf9832b1f4ca1c8075a1035405290856a0a182e8cf51fc6895753",
      "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."
  }
}