{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261004-advance-20261005-r03-bijective-0097",
  "pilot_number": "R0004-02822",
  "number": 2822,
  "family_id": "qf-bijective-cardinality-possibility",
  "title": {
    "zh-Hant": "第 2822 題：有限基數下的雙射存在可能性",
    "en": "Question 2822: Existence of a Bijection from Finite Cardinalities"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261004-advance-20261005-r03-bijective-0097",
    "batch_id": "batch-20261004-advance-20261005-r03-bijective-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-bijective"
    ],
    "title_zh": "有限基數下的雙射存在可能性",
    "stem_zh": "明示有限基數 |A|=10、|B|=11，未提供任何特定函數表。是否存在某個 A→B 的雙射？",
    "answer": {
      "kind": "short_text",
      "value": "否",
      "explanation_zh": "|A|=10；|B|=11。有限集合可逐一配對恰在基數相等時成立；這不是判斷給定函數，零到零也包含唯一空雙射。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 3,
      "skills": [
        "有限基數下的雙射存在可能性"
      ]
    },
    "variant": {
      "family_id": "qf-bijective-cardinality-possibility",
      "seed": 202610053097,
      "parameters": {
        "a_size": 10,
        "b_size": 11,
        "operation": "cardinality_possibility",
        "semantic_template": "qf-bijective-cardinality-possibility/{\"a_size\":10,\"b_size\":11,\"operation\":\"cardinality_possibility\"}"
      }
    },
    "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": "Codex independent producer (inherited model; exact runtime model not exposed)",
      "generated_at": "2026-10-04",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261004-advance-20261005-r03-bijective-0097",
    "batch_id": "batch-20261004-advance-20261005-r03-bijective-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "75e0b00bdb1b9776a8bdeec88d312d2f3bb06cc1185e38b65ab0aa1e431b999b",
      "family_id": "qf-bijective-cardinality-possibility",
      "seed": 202610053097,
      "semantic_template": "qf-bijective-cardinality-possibility/{\"a_size\":10,\"b_size\":11,\"operation\":\"cardinality_possibility\"}"
    },
    "fields": {
      "title": "Existence of a Bijection from Finite Cardinalities",
      "stem": "The explicit finite cardinalities are |A|=10 and |B|=11, with no particular function table supplied. Does some bijection A→B exist?",
      "answer": {
        "kind": "short_text",
        "value": "No",
        "explanation": "Finite sets admit a bijection exactly when their cardinalities agree, giving No. This asks about existence rather than testing a given function; two empty sets admit the unique empty bijection."
      },
      "skills": [
        "Existence of a Bijection from Finite Cardinalities"
      ],
      "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": "明示有限基數 |A|=10、|B|=11，未提供任何特定函數表。是否存在某個 A→B 的雙射？",
        "en": "The explicit finite cardinalities are |A|=10 and |B|=11, with no particular function table supplied. Does some bijection A→B exist?"
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "finite |A|=10; |B|=11",
      "after": "equal finite cardinalities=否",
      "relation": "implies",
      "rule_id": "finite_bijection_cardinality",
      "conditions": [],
      "explanation": {
        "zh-Hant": "有限集合間存在雙射當且僅當元素數相等。雙射把每個A元素配給一個不同B元素且不遺漏任何B元素，故兩個數相等。",
        "en": "A bijection between finite sets exists exactly when their cardinalities agree.A bijection pairs each A-element with a distinct B-element and misses no B-element,so the counts agree."
      },
      "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": "m=10; n=11",
      "after": "equal sizes:pair indexed elements;unequal sizes:no bijection",
      "relation": "implies",
      "rule_id": "finite_pairing_construction",
      "conditions": [],
      "explanation": {
        "zh-Hant": "相等時任選有限列舉並按索引逐一配對就是雙射；不等時完整的一對一配對不可能。0到0是唯一空函數，亦為雙射。",
        "en": "For equal sizes,enumerate both finite sets and pair equal indices to construct a bijection.For unequal sizes a complete one-to-one pairing is impossible.The unique empty function from0 to0 is a bijection."
      },
      "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": "existence of some map,without any supplied table",
      "after": "not a verdict about an unspecified particular function",
      "relation": "implies",
      "rule_id": "existence_vs_given_function",
      "conditions": [],
      "explanation": {
        "zh-Hant": "即使基數相等，也不能推斷任意給定函數必為雙射；本題只問某個雙射是否存在。",
        "en": "Equal cardinalities do not make every particular function bijective;this task asks only whether some bijection exists."
      },
      "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": false,
    "claim_scope": "existence_only"
  },
  "conclusion": {
    "canonical_value": "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-20261004-advance-20261005-r03-bijective-foundations",
      "batch_path": "public/data/questions/batches/batch-20261004-advance-20261005-r03-bijective-foundations/questions.jsonl",
      "batch_sha256": "f4a605cc0336af363b5bdd962e1eedcd3f9df30552bca14e1231a63e56c4783e",
      "original_line": 97,
      "original_line_sha256": "3042078dd4aa4cd864dddfca747ac10a16bcf859380341d8e77a4033476bb19a",
      "question_compact_sha256": "75e0b00bdb1b9776a8bdeec88d312d2f3bb06cc1185e38b65ab0aa1e431b999b",
      "english_row_compact_sha256": "e8f473609e36155140701d0f0c206fbb8f78af4d9e139cea9b6c35283a0d0a4b",
      "english_manifest_path": "public/data/i18n/editions/batch-20261004-advance-20261005-r03-bijective-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261004-advance-20261005-r03-bijective-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-bijective": {
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        "english_sha256": "6263c10144a28f0b03f69a5c27ee3b32c7eaf33f0446b48924c55a2d093b71e9",
        "original_path": "public/data/mko/bijective.json",
        "english_path": "public/data/i18n/editions/mko-bijective.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "b254245aa10315455f756d4620a6ac0243e0146bccea5d89e872f1359bb32a1d",
    "original_candidate_sha256": "d95f0bbb5d94f9ea9941acaef074f3cbd81c2be405a8c761ff908685cf734970",
    "original_candidate_line_sha256": "d95f0bbb5d94f9ea9941acaef074f3cbd81c2be405a8c761ff908685cf734970"
  },
  "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-p5",
    "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": "312ae86315346b8955842a547cb2d09246e1f826755f72c2a51f042e5c7d5c56",
      "question_first_witness_sha256": "831f449ed41ff9144a290e534aa32adf7252d15b81c16ed5886a7dfd65dee70c"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 3,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "89073105a4efa80b0ffb4f757f6d197a533989dfa82dbd78c0e4f63f0614512e",
      "arithmetic_steps": 0
    },
    "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."
  }
}