{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261008-advance-20261011-r07-tends-to-relation-0053",
  "pilot_number": "R0009-05203",
  "number": 5203,
  "family_id": "qf-density11-tends-principal-candidate-mask",
  "title": {
    "zh-Hant": "第 5203 題：有限主濾子的反例候選資格",
    "en": "Question 5203: Typed Principal-Filter Obstruction Candidates"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261008-advance-20261011-r07-tends-to-relation-0053",
    "batch_id": "batch-20261008-advance-20261011-r07-tends-to-relation-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-tends-to-relation"
    ],
    "title_zh": "有限主濾子的反例候選資格",
    "stem_zh": "有限主濾子；完整總映射 f:Fin(6)→Fin(2) 的表=[0, 1, 1, 0, 0, 1]，S=[0, 1, 2]；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。T=[0]；目標principal(T)由包含T的所有目標子集構成；有序候選=[0, 0, 0]。逐項答是否同時c∈S且f(c)∉T；型別外候選不求f(c)。",
    "answer": {
      "kind": "expression",
      "value": "[否, 否, 否]",
      "explanation_zh": "主濾子反例候選資格=[否, 否, 否]。先檢型別與S資格才讀映射值，未屬S及映入T者都不是阻擋；非一般數列或點極限。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 1,
      "skills": [
        "有限主濾子的反例候選資格"
      ]
    },
    "variant": {
      "family_id": "qf-density11-tends-principal-candidate-mask",
      "seed": 202610117053,
      "parameters": {
        "n": 6,
        "m": 2,
        "images": [
          0,
          1,
          1,
          0,
          0,
          1
        ],
        "S": [
          0,
          1,
          2
        ],
        "T": [
          0
        ],
        "candidates": [
          0,
          0,
          0
        ],
        "filter_model": "principal_generators",
        "operation": "density11_tends_principal_candidate_mask",
        "domain": "finite_principal_filter_model",
        "semantic_template": "qf-density11-tends-principal-candidate-mask/[\"maps_inside\",\"maps_inside\",\"maps_inside\"]"
      }
    },
    "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-0053",
    "batch_id": "batch-20261008-advance-20261011-r07-tends-to-relation-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "4fb73b667dd6ec926f072551fa64d230659c604dd5e3d31b0a795432a8aa45ff",
      "family_id": "qf-density11-tends-principal-candidate-mask",
      "seed": 202610117053,
      "semantic_template": "qf-density11-tends-principal-candidate-mask/[\"maps_inside\",\"maps_inside\",\"maps_inside\"]"
    },
    "fields": {
      "title": "Typed Principal-Filter Obstruction Candidates",
      "stem": "The complete total map f:Fin(6)→Fin(2) has images=[0, 1, 1, 0, 0, 1] indexed from 0; Fin(6)={0,…,5}. Every table entry must lie in the target type. S=[0, 1, 2]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0], ordered candidates=[0, 0, 0]. For each c, answer whether c∈S and f(c)∉T; do not evaluate f at a candidate outside its type.",
      "answer": {
        "kind": "expression",
        "value": "[No, No, No]",
        "explanation": "Check source typing and S membership before reading f; outside-S candidates and images inside T are not obstructions. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [No, No, No]."
      },
      "skills": [
        "Typed Principal-Filter Obstruction Candidates"
      ],
      "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(6)→Fin(2) 的表=[0, 1, 1, 0, 0, 1]，S=[0, 1, 2]；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。T=[0]；目標principal(T)由包含T的所有目標子集構成；有序候選=[0, 0, 0]。逐項答是否同時c∈S且f(c)∉T；型別外候選不求f(c)。",
        "en": "The complete total map f:Fin(6)→Fin(2) has images=[0, 1, 1, 0, 0, 1] indexed from 0; Fin(6)={0,…,5}. Every table entry must lie in the target type. S=[0, 1, 2]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0], ordered candidates=[0, 0, 0]. For each c, answer whether c∈S and f(c)∉T; do not evaluate f at a candidate outside its type."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source index=0∈Fin(6)",
      "after": "image=0∈Fin(2)",
      "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(6)",
      "after": "image=1∈Fin(2)",
      "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(6)",
      "after": "image=1∈Fin(2)",
      "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(6)",
      "after": "image=0∈Fin(2)",
      "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(6)",
      "after": "image=0∈Fin(2)",
      "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(6)",
      "after": "image=1∈Fin(2)",
      "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": "Fin(6)→Fin(2);S=[0, 1, 2];full map=[0, 1, 1, 0, 0, 1]",
      "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": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "candidate=0;inside Fin(n)=是;inside S=是",
      "after": "image if in S=0;obstruction=否",
      "relation": "implies",
      "rule_id": "typed_principal_candidate_obstruction",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先檢型別及S資格；不屬S時不讀表，避免負索引或域外值冒充合法映射求值。",
        "en": "Check type and S-membership before table access;outside S do not read the table,so negative or outside indices cannot masquerade as legal map evaluations."
      },
      "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": "candidate=0;inside Fin(n)=是;inside S=是",
      "after": "image if in S=0;obstruction=否",
      "relation": "implies",
      "rule_id": "typed_principal_candidate_obstruction",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先檢型別及S資格；不屬S時不讀表，避免負索引或域外值冒充合法映射求值。",
        "en": "Check type and S-membership before table access;outside S do not read the table,so negative or outside indices cannot masquerade as legal map evaluations."
      },
      "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": "candidate=0;inside Fin(n)=是;inside S=是",
      "after": "image if in S=0;obstruction=否",
      "relation": "implies",
      "rule_id": "typed_principal_candidate_obstruction",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先檢型別及S資格；不屬S時不讀表，避免負索引或域外值冒充合法映射求值。",
        "en": "Check type and S-membership before table access;outside S do not read the table,so negative or outside indices cannot masquerade as legal map evaluations."
      },
      "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,
      false,
      false
    ],
    "candidate_rows": [
      [
        0,
        true,
        true,
        0,
        false
      ],
      [
        0,
        true,
        true,
        0,
        false
      ],
      [
        0,
        true,
        true,
        0,
        false
      ]
    ]
  },
  "conclusion": {
    "canonical_value": "[否, 否, 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-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": 53,
      "original_line_sha256": "ad9bc912a51845f52d937749a24522b5356b62e2b246cf3f40fbbf668895cc7d",
      "question_compact_sha256": "4fb73b667dd6ec926f072551fa64d230659c604dd5e3d31b0a795432a8aa45ff",
      "english_row_compact_sha256": "709659146ebc15ee33cbcf586f02e91cf29c5b98fd39d63a4363440d2796b847",
      "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": "c51a97f30cdbcd40e09a393e9d31ead553e1bf9dd934a1ea0d67fca054a7dd25",
    "original_candidate_line_sha256": "c51a97f30cdbcd40e09a393e9d31ead553e1bf9dd934a1ea0d67fca054a7dd25"
  },
  "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",
      "question_first_witness_sha256": "5a43f4f81b6256281f8d139a4010bbde127be803c528e5878857269c8f854462"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 10,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "e59a33a00aaa904e921f10d1d1db1aef8ff8c0abe34674d555a2881b89b9ac2e",
      "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."
  }
}