{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261008-advance-20261011-r02-arithmetic-operations-0089",
  "pilot_number": "R0008-04689",
  "number": 4689,
  "family_id": "qf-density11-arithmetic-system-membership",
  "title": {
    "zh-Hant": "第 4689 題：Q運算結果與自然數封閉資格",
    "en": "Question 4689: Q Results and Natural-System Eligibility"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261008-advance-20261011-r02-arithmetic-operations-0089",
    "batch_id": "batch-20261008-advance-20261011-r02-arithmetic-operations-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-arithmetic-operations"
    ],
    "title_zh": "Q運算結果與自然數封閉資格",
    "stem_zh": "有理數Q運算，N包含0；自然數輸入a=10、b=4。依序求 (a−b, 差是否屬N, a/b, 商是否屬N)；商或其資格未定義則寫未定義。",
    "answer": {
      "kind": "expression",
      "value": "(6, 是, 5/2, 否)",
      "explanation_zh": "Q結果與N封閉資格=(6, 是, 5/2, 否)。使用精確Q結果，非截斷減法或整除；負差不屬N，非整數商不屬N，除零未定義。"
    },
    "difficulty": {
      "band": "lower_secondary",
      "level": 2,
      "skills": [
        "Q運算結果與自然數封閉資格"
      ]
    },
    "variant": {
      "family_id": "qf-density11-arithmetic-system-membership",
      "seed": 202610112089,
      "parameters": {
        "a": 10,
        "b": 4,
        "operation": "density11_arithmetic_system_membership",
        "domain": "N_operands_Q_results",
        "semantic_template": "qf-density11-arithmetic-system-membership/[10,4]"
      }
    },
    "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-0089",
    "batch_id": "batch-20261008-advance-20261011-r02-arithmetic-operations-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "23fa51b0222cc6326c09fcfe596870a4f52d7cf5e68534f978c7d60630760b27",
      "family_id": "qf-density11-arithmetic-system-membership",
      "seed": 202610112089,
      "semantic_template": "qf-density11-arithmetic-system-membership/[10,4]"
    },
    "fields": {
      "title": "Q Results and Natural-System Eligibility",
      "stem": "All operations and evaluations are exact in Q, not truncated natural-number arithmetic. N includes 0. Natural inputs a=10, b=4. Give (a−b, does the difference belong to N?, a/b, does the quotient belong to N?). Use undefined for an undefined quotient or its eligibility.",
      "answer": {
        "kind": "expression",
        "value": "(6, Yes, 5/2, No)",
        "explanation": "A negative difference is outside N and a noninteger quotient is outside N; neither truncated subtraction nor integer division is used, and division by zero stays undefined. The exact result, retaining the requested order, is (6, Yes, 5/2, No)."
      },
      "skills": [
        "Q Results and Natural-System Eligibility"
      ],
      "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運算，N包含0；自然數輸入a=10、b=4。依序求 (a−b, 差是否屬N, a/b, 商是否屬N)；商或其資格未定義則寫未定義。",
        "en": "All operations and evaluations are exact in Q, not truncated natural-number arithmetic. N includes 0. Natural inputs a=10, b=4. Give (a−b, does the difference belong to N?, a/b, does the quotient belong to N?). Use undefined for an undefined quotient or its eligibility."
      }
    }
  ],
  "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": "Q difference := sub(10, 4)",
      "after": "Q difference = 6",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "在Q中相減，負結果也保留。",
        "en": "Subtract in Q,retaining a negative result."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Q difference，不以小數近似替代。",
          "en": "Recompute Q difference with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "10",
          "4"
        ],
        "output": "6",
        "label": "Q difference"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "Q quotient := div(10, 4)",
      "after": "Q quotient = 5/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "分母非零才做精確除法。",
        "en": "Use exact division only with a nonzero denominator."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Q quotient，不以小數近似替代。",
          "en": "Recompute Q quotient with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "10",
          "4"
        ],
        "output": "5/2",
        "label": "Q quotient"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "a=10;b=4;difference=6",
      "after": "(6, 是, 5/2, 否)",
      "relation": "implies",
      "rule_id": "rational_result_natural_membership",
      "conditions": [],
      "explanation": {
        "zh-Hant": "屬N要求非負且整數；未定義的商沒有可供判定N資格的數值，不能把資格填成否或是。",
        "en": "Membership in N requires a nonnegative integer.An undefined quotient has no numeric N-membership verdict,so do not fill its verdict with Yes or No."
      },
      "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": [
      6,
      true,
      "5/2",
      false
    ]
  },
  "conclusion": {
    "canonical_value": "(6, 是, 5/2, 否)",
    "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": 89,
      "original_line_sha256": "e6f2647e2582d12200e8e1a4b9fb6ec506b369a006f270a85bee66d683b702cd",
      "question_compact_sha256": "23fa51b0222cc6326c09fcfe596870a4f52d7cf5e68534f978c7d60630760b27",
      "english_row_compact_sha256": "0a60c4aeb3a0a0d9fb2e003af23303fdbfe45008d86e809bec8572830177e82f",
      "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"
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    "mko_bindings": {
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        "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": "cb9ad26f4dac0e7f7b57fce687057c867ab9765a17775f0e9b3a15531e9a162f",
    "original_candidate_line_sha256": "cb9ad26f4dac0e7f7b57fce687057c867ab9765a17775f0e9b3a15531e9a162f"
  },
  "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": "63a7404acc86d1d825726b7a408a9149f64589966ce54c8523d881aa833a7826",
      "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."
  }
}