{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261014-advance-20261030-r05-cardinality-comparison-0030",
  "pilot_number": "R0028-14880",
  "number": 14880,
  "family_id": "qf-mod30-cardinality-surjection-finite",
  "title": {
    "zh-Hant": "第 14880 題：集合基數比較",
    "en": "Question 14880: A Surjection Supplies the Reverse Comparison Only in This Finite Scope"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261014-advance-20261030-r05-cardinality-comparison-0030",
    "batch_id": "batch-20261014-advance-20261030-r05-cardinality-comparison-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-cardinality-comparison"
    ],
    "title_zh": "集合基數比較",
    "stem_zh": "mapping綁精確有限source與target。操作=cardinality-surjection-finite；資料={\"source\":[40,41],\"target\":[140,141,142],\"mapping\":[140,141]}。請依 [source,target,mapping,surjective,comparison,boundary] 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "([40, 41], [140, 141, 142], [140, 141], 否, no_comparison_certificate, finite_only)",
      "explanation_zh": "Wave4第四層核對=([40, 41], [140, 141, 142], [140, 141], 否, no_comparison_certificate, finite_only)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；injective surjective bijective責任分開；surjection逆向比較限有限集合且不外推無限。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 1,
      "skills": [
        "集合基數比較"
      ]
    },
    "variant": {
      "family_id": "qf-mod30-cardinality-surjection-finite",
      "seed": 202610305030,
      "parameters": {
        "source": [
          40,
          41
        ],
        "target": [
          140,
          141,
          142
        ],
        "mapping": [
          140,
          141
        ],
        "domain": "Exact_finite_cardinality_mapping",
        "operation": "cardinality-surjection-finite",
        "semantic_template": "qf-mod30-cardinality-surjection-finite/{\"mapping\":[140,141],\"source\":[40,41],\"target\":[140,141,142]}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_finite_model",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-14",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261014-advance-20261030-r05-cardinality-comparison-0030",
    "batch_id": "batch-20261014-advance-20261030-r05-cardinality-comparison-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "c5dc03060e7322041ae7a897575c2540b20de0c30ad1af3d27052697c5b3bbbb",
      "family_id": "qf-mod30-cardinality-surjection-finite",
      "seed": 202610305030,
      "semantic_template": "qf-mod30-cardinality-surjection-finite/{\"mapping\":[140,141],\"source\":[40,41],\"target\":[140,141,142]}"
    },
    "fields": {
      "title": "A Surjection Supplies the Reverse Comparison Only in This Finite Scope",
      "stem": "Use the complete finite source=[40, 41] and target=[140, 141, 142] as distinct-element lists in their original orders, with mapping=[140, 141] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,surjective?,comparison,finite_only). comparison is |source|≥|target| when this mapping is surjective, otherwise no_comparison_certificate.",
      "answer": {
        "kind": "expression",
        "value": "([40, 41], [140, 141, 142], [140, 141], No, no_comparison_certificate, finite_only)",
        "explanation": "Test coverage of every target element by the specified mapping. The reverse finite-cardinality comparison is a certificate from this mapping, not an unrestricted assertion about infinite sets or choice. The exact result, retaining the requested order, is ([40, 41], [140, 141, 142], [140, 141], No, no_comparison_certificate, finite_only)."
      },
      "skills": [
        "A Surjection Supplies the Reverse Comparison Only in This Finite Scope"
      ],
      "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": "mapping綁精確有限source與target。操作=cardinality-surjection-finite；資料={\"source\":[40,41],\"target\":[140,141,142],\"mapping\":[140,141]}。請依 [source,target,mapping,surjective,comparison,boundary] 欄序回答。",
        "en": "Use the complete finite source=[40, 41] and target=[140, 141, 142] as distinct-element lists in their original orders, with mapping=[140, 141] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,surjective?,comparison,finite_only). comparison is |source|≥|target| when this mapping is surjective, otherwise no_comparison_certificate."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "exact ordered source=[40, 41];exact ordered target=[140, 141, 142];aligned mapping=[140, 141]",
      "after": "total typed map on these distinct source/target lists;orders retained",
      "relation": "implies",
      "rule_id": "cardinality_exact_bound_total_mapping",
      "conditions": [],
      "explanation": {
        "zh-Hant": "除pigeonhole集合計數外，給定映射須綁原序且無重複的source/target，對每個來源恰有合法目標值；不只看清單長度。",
        "en": "A supplied mapping binds distinct source/target lists in original order,with exactly one permitted image for each source element.Length comparison alone is not mapping qualification."
      },
      "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": "distinct source indices=[0, 1];source labels=[40, 41];images=[140, 141]",
      "after": "same image=否",
      "relation": "implies",
      "rule_id": "cardinality_every_source_pair_collision_check",
      "conditions": [],
      "explanation": {
        "zh-Hant": "來源元素已確認不同，逐對核像是否相同；空來源單射條件空泛通過，這本身不給滿射。",
        "en": "The source elements are distinct;compare images for every pair.An empty source is vacuously injective,without automatically being surjective."
      },
      "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": "target label=140;exact original mapping=[140, 141]",
      "after": "preimage source indices=[0];covered=是",
      "relation": "implies",
      "rule_id": "cardinality_every_target_fiber_coverage",
      "conditions": [],
      "explanation": {
        "zh-Hant": "目標每個元素都需非空原像才滿射；原像記原來源索引，不把像值當索引。空目標的覆蓋責任空泛成立，但總映射合型前提仍須先通過。",
        "en": "Every target element needs a nonempty preimage for surjectivity.Record original source indices,not image-as-index values.Coverage of an empty target is vacuous,but total-map typedness still must qualify first."
      },
      "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": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "target label=141;exact original mapping=[140, 141]",
      "after": "preimage source indices=[1];covered=是",
      "relation": "implies",
      "rule_id": "cardinality_every_target_fiber_coverage",
      "conditions": [],
      "explanation": {
        "zh-Hant": "目標每個元素都需非空原像才滿射；原像記原來源索引，不把像值當索引。空目標的覆蓋責任空泛成立，但總映射合型前提仍須先通過。",
        "en": "Every target element needs a nonempty preimage for surjectivity.Record original source indices,not image-as-index values.Coverage of an empty target is vacuous,but total-map typedness still must qualify first."
      },
      "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": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "target label=142;exact original mapping=[140, 141]",
      "after": "preimage source indices=[];covered=否",
      "relation": "implies",
      "rule_id": "cardinality_every_target_fiber_coverage",
      "conditions": [],
      "explanation": {
        "zh-Hant": "目標每個元素都需非空原像才滿射；原像記原來源索引，不把像值當索引。空目標的覆蓋責任空泛成立，但總映射合型前提仍須先通過。",
        "en": "Every target element needs a nonempty preimage for surjectivity.Record original source indices,not image-as-index values.Coverage of an empty target is vacuous,but total-map typedness still must qualify first."
      },
      "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": "surjective=否;source size=2;target size=3",
      "after": "this mapping reverse comparison certificate=no_comparison_certificate;finite_only",
      "relation": "implies",
      "rule_id": "surjection_finite_scope_and_failure_certificate",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐目標覆蓋與有限原像選取支撐此比較；No只表示本映射未給證書，不是把其他集合關係或無限版本一併否定。",
        "en": "Target coverage and finite preimage selection support this comparison.No means this mapping supplied no certificate,not a denial of other set relations or an infinite version."
      },
      "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": [
      [
        40,
        41
      ],
      [
        140,
        141,
        142
      ],
      [
        140,
        141
      ],
      false,
      "no_comparison_certificate",
      "finite_only"
    ]
  },
  "conclusion": {
    "canonical_value": "([40, 41], [140, 141, 142], [140, 141], 否, no_comparison_certificate, finite_only)",
    "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-20261014-advance-20261030-r05-cardinality-comparison-foundations",
      "batch_path": "public/data/questions/batches/batch-20261014-advance-20261030-r05-cardinality-comparison-foundations/questions.jsonl",
      "batch_sha256": "601304f1f2812ae71284fc9774945ccecf61443483ed83e7496eeb8e794c970f",
      "original_line": 30,
      "original_line_sha256": "ddd750d183453090053d9cfdf13502387c001cc1532d5f85c848d2822d85f6db",
      "question_compact_sha256": "c5dc03060e7322041ae7a897575c2540b20de0c30ad1af3d27052697c5b3bbbb",
      "english_row_compact_sha256": "d28c8391ad5c7e40d765228cf2ffea912e95301f757dd196fd3404c3d74579e9",
      "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261030-r05-cardinality-comparison-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261030-r05-cardinality-comparison-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-cardinality-comparison": {
        "sha256": "50efb74b57a576c8cec0b8230fc37bf7cd9c7ecd6303efe09810085d9a570227",
        "english_sha256": "d5d927d20ab09e9e7df6d3a33e4ed265b08e1db823fee7f5cb2a61a667172b4a",
        "original_path": "public/data/mko/cardinality-comparison.json",
        "english_path": "public/data/i18n/editions/mko-cardinality-comparison.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "466f72389eb454e24322e59df2871ec206c351545a4a7b072d39fb5f6afc2697",
    "original_candidate_sha256": "7784236acb32693de7cc8bd5a4a4f0b55e697c90b71424761534257001ad40cc",
    "original_candidate_line_sha256": "7784236acb32693de7cc8bd5a4a4f0b55e697c90b71424761534257001ad40cc"
  },
  "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-20261026-007-p29",
    "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": "bfea39236e49aa7e44aa18bfa0d7857ff34a6acb762d0599e20b0f5af0f209ae",
      "question_first_witness_sha256": "61c433f5c1e55ed4d84d08557fb8c117d26bbd404fdf05519ad42c09324c36e0"
    },
    "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": "86d86fbf21629de166f5528f854ff00236d0dd3f273de42d63b309f20092e270",
      "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."
  }
}