{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261008-advance-20261011-r07-tends-to-relation-0022",
  "pilot_number": "R0009-05272",
  "number": 5272,
  "family_id": "qf-density11-tends-principal-image",
  "title": {
    "zh-Hant": "第 5272 題：有限主濾子的像生成集",
    "en": "Question 5272: Finite Principal-Filter Image Generator"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261008-advance-20261011-r07-tends-to-relation-0022",
    "batch_id": "batch-20261008-advance-20261011-r07-tends-to-relation-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-tends-to-relation"
    ],
    "title_zh": "有限主濾子的像生成集",
    "stem_zh": "有限主濾子；完整總映射 f:Fin(8)→Fin(8) 的表=[0, 1, 2, 3, 4, 5, 6, 7]，S=[0, 2, 4]；principal(S)由包含S的所有來源子集構成，允許空生成集且不加入NeBot。列像主濾子的生成集f(S)，遞增且去重。",
    "answer": {
      "kind": "expression",
      "value": "[0, 2, 4]",
      "explanation_zh": "主濾子像生成集=[0, 2, 4]。只把S的每個元素送入合法總映射，再去重；空生成集的像仍空，非一般數列或點極限。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 3,
      "skills": [
        "有限主濾子的像生成集"
      ]
    },
    "variant": {
      "family_id": "qf-density11-tends-principal-image",
      "seed": 202610117022,
      "parameters": {
        "n": 8,
        "m": 8,
        "images": [
          0,
          1,
          2,
          3,
          4,
          5,
          6,
          7
        ],
        "S": [
          0,
          2,
          4
        ],
        "filter_model": "principal_generators",
        "operation": "density11_tends_principal_image",
        "domain": "finite_principal_filter_model",
        "semantic_template": "qf-density11-tends-principal-image/[0,2,4]"
      }
    },
    "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-r07-tends-to-relation-0022",
    "batch_id": "batch-20261008-advance-20261011-r07-tends-to-relation-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "a181049245d7d8ee1b4b4fcb81167c15a10e39ffd86e8960114add7690a63b39",
      "family_id": "qf-density11-tends-principal-image",
      "seed": 202610117022,
      "semantic_template": "qf-density11-tends-principal-image/[0,2,4]"
    },
    "fields": {
      "title": "Finite Principal-Filter Image Generator",
      "stem": "The complete total map f:Fin(8)→Fin(8) has images=[0, 1, 2, 3, 4, 5, 6, 7] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 2, 4]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. List f(S) as an increasing set without repetition.",
      "answer": {
        "kind": "expression",
        "value": "[0, 2, 4]",
        "explanation": "Map exactly the elements of S through the legal total map and deduplicate; the empty generator still has an empty image. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [0, 2, 4]."
      },
      "skills": [
        "Finite Principal-Filter Image Generator"
      ],
      "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": "有限主濾子；完整總映射 f:Fin(8)→Fin(8) 的表=[0, 1, 2, 3, 4, 5, 6, 7]，S=[0, 2, 4]；principal(S)由包含S的所有來源子集構成，允許空生成集且不加入NeBot。列像主濾子的生成集f(S)，遞增且去重。",
        "en": "The complete total map f:Fin(8)→Fin(8) has images=[0, 1, 2, 3, 4, 5, 6, 7] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 2, 4]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. List f(S) as an increasing set without repetition."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source index=0∈Fin(8)",
      "after": "image=0∈Fin(8)",
      "relation": "implies",
      "rule_id": "complete_principal_map_row",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整表的每項都要合法，不能只因稍後未使用某項就略過總映射前提。",
        "en": "Every entry of the complete table must be legal;an entry being unused later does not waive total-map typing."
      },
      "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": "source index=1∈Fin(8)",
      "after": "image=1∈Fin(8)",
      "relation": "implies",
      "rule_id": "complete_principal_map_row",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整表的每項都要合法，不能只因稍後未使用某項就略過總映射前提。",
        "en": "Every entry of the complete table must be legal;an entry being unused later does not waive total-map typing."
      },
      "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": "source index=2∈Fin(8)",
      "after": "image=2∈Fin(8)",
      "relation": "implies",
      "rule_id": "complete_principal_map_row",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整表的每項都要合法，不能只因稍後未使用某項就略過總映射前提。",
        "en": "Every entry of the complete table must be legal;an entry being unused later does not waive total-map typing."
      },
      "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": "source index=3∈Fin(8)",
      "after": "image=3∈Fin(8)",
      "relation": "implies",
      "rule_id": "complete_principal_map_row",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整表的每項都要合法，不能只因稍後未使用某項就略過總映射前提。",
        "en": "Every entry of the complete table must be legal;an entry being unused later does not waive total-map typing."
      },
      "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": "source index=4∈Fin(8)",
      "after": "image=4∈Fin(8)",
      "relation": "implies",
      "rule_id": "complete_principal_map_row",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整表的每項都要合法，不能只因稍後未使用某項就略過總映射前提。",
        "en": "Every entry of the complete table must be legal;an entry being unused later does not waive total-map typing."
      },
      "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": "source index=5∈Fin(8)",
      "after": "image=5∈Fin(8)",
      "relation": "implies",
      "rule_id": "complete_principal_map_row",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整表的每項都要合法，不能只因稍後未使用某項就略過總映射前提。",
        "en": "Every entry of the complete table must be legal;an entry being unused later does not waive total-map typing."
      },
      "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": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "source index=6∈Fin(8)",
      "after": "image=6∈Fin(8)",
      "relation": "implies",
      "rule_id": "complete_principal_map_row",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整表的每項都要合法，不能只因稍後未使用某項就略過總映射前提。",
        "en": "Every entry of the complete table must be legal;an entry being unused later does not waive total-map typing."
      },
      "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": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "source index=7∈Fin(8)",
      "after": "image=7∈Fin(8)",
      "relation": "implies",
      "rule_id": "complete_principal_map_row",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整表的每項都要合法，不能只因稍後未使用某項就略過總映射前提。",
        "en": "Every entry of the complete table must be legal;an entry being unused later does not waive total-map typing."
      },
      "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": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "Fin(8)→Fin(8);S=[0, 2, 4];full map=[0, 1, 2, 3, 4, 5, 6, 7]",
      "after": "principal(S)={A:S⊆A};empty generators allowed,no NeBot",
      "relation": "implies",
      "rule_id": "finite_principal_generator_convention",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依明示約定允許S空；此時包含S的族含空集，不另加真濾子或NeBot條件。這是有限主生成集模型，不是假定一般數列或點極限。",
        "en": "The declared convention allows empty S;its family of supersets then contains the empty set,and no proper-filter or NeBot condition is added.This is a finite principal-generator model,not a general sequence or point limit."
      },
      "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": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "source generator x=0∈S",
      "after": "f(x)=0",
      "relation": "implies",
      "rule_id": "source_generator_image_row",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只取S內的來源元素作像生成集，不把全部陪域誤作f(S)。",
        "en": "Map only source generators in S;do not confuse the whole codomain with f(S)."
      },
      "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": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "source generator x=2∈S",
      "after": "f(x)=2",
      "relation": "implies",
      "rule_id": "source_generator_image_row",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只取S內的來源元素作像生成集，不把全部陪域誤作f(S)。",
        "en": "Map only source generators in S;do not confuse the whole codomain with f(S)."
      },
      "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": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "source generator x=4∈S",
      "after": "f(x)=4",
      "relation": "implies",
      "rule_id": "source_generator_image_row",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只取S內的來源元素作像生成集，不把全部陪域誤作f(S)。",
        "en": "Map only source generators in S;do not confuse the whole codomain with f(S)."
      },
      "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": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "[0, 2, 4]",
      "after": "A⊇f(S) iff f⁻¹(A)⊇S",
      "relation": "implies",
      "rule_id": "principal_map_generator_identity",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依主生成族定義，包含f(S)等價於逆像包含S，故像生成集正是f(S)；重像只列一次。",
        "en": "By the principal-family definition,containing f(S) is equivalent to the preimage containing S,so f(S) is the image generator.Duplicate image values appear once."
      },
      "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": [
      0,
      2,
      4
    ]
  },
  "conclusion": {
    "canonical_value": "[0, 2, 4]",
    "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-r07-tends-to-relation-foundations",
      "batch_path": "public/data/questions/batches/batch-20261008-advance-20261011-r07-tends-to-relation-foundations/questions.jsonl",
      "batch_sha256": "3ca90336b05f08a7567b56044185bef7fc4f4f8286e5afeac67e950f7c74bfde",
      "original_line": 22,
      "original_line_sha256": "6610c8422c5813173d263e804af4091ac319bedb12d58866f824a0576a6c0b74",
      "question_compact_sha256": "a181049245d7d8ee1b4b4fcb81167c15a10e39ffd86e8960114add7690a63b39",
      "english_row_compact_sha256": "b79eef40e7ab49b2817944579149e437b09763d943fb9e6a7ddc29df2625c216",
      "english_manifest_path": "public/data/i18n/editions/batch-20261008-advance-20261011-r07-tends-to-relation-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261008-advance-20261011-r07-tends-to-relation-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-tends-to-relation": {
        "sha256": "2fbb3c700378e80e68833bcada18866ed6439803d8b639e304fe76d31903f70e",
        "english_sha256": "bed5d2c178f726aeda66b8705697852f39ce93b2d1a6807b7acbb3c97dfd4ac4",
        "original_path": "public/data/mko/tends-to-relation.json",
        "english_path": "public/data/i18n/editions/mko-tends-to-relation.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "3132b1200be7a5de38191badc52fb1727d4dd9d4c2f26818de2e8a76426d2d3f",
    "original_candidate_sha256": "d79529dcec7fc4d0aee9665d8ede9f2dd783fd0b0c57740afb16beca1b6ef1ca",
    "original_candidate_line_sha256": "d79529dcec7fc4d0aee9665d8ede9f2dd783fd0b0c57740afb16beca1b6ef1ca"
  },
  "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-20261026-005-p10",
    "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": "71e7db519df31f3a164533e472e293cb64f001b475a8d002b4a5c2740a237b56",
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    "bilingual_semantics": true,
    "steps_reviewed": 13,
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    "authority": "Owner-authorized reviewed reader publication; original question/candidate states remain historical and unchanged."
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}