{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261009-advance-20261013-r06-event-0072",
  "pilot_number": "R0011-06097",
  "number": 6097,
  "family_id": "qf-density13-event-partition-family-qualification",
  "title": {
    "zh-Hant": "第 6097 題：分塊聯集族的資格與最小閉包",
    "en": "Question 6097: Partition-Union Event Eligibility and Least Containing Event"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261009-advance-20261013-r06-event-0072",
    "batch_id": "batch-20261009-advance-20261013-r06-event-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-event"
    ],
    "title_zh": "分塊聯集族的資格與最小閉包",
    "stem_zh": "S=Fin(n)，n=8；F為完整分塊的聯集族；完整互斥非空blocks=[[0, 3, 4], [1], [2], [5], [6], [7]]，候選E=[0]⊆S。依序答 (E是否完整分塊聯集, 部分命中的分塊成員列表, 最小包含E的F集合)，塊內排序、塊外按數值字典序。",
    "answer": {
      "kind": "expression",
      "value": "(否, [[0, 3, 4]], [0, 3, 4])",
      "explanation_zh": "分塊事件資格與閉包=(否, [[0, 3, 4]], [0, 3, 4])。每塊只准全取或全不取；閉包取所有有命中的整塊，整塊命中不列部分；有限S不自動等於F=P(S)，集合資料不給概率或獨立性。"
    },
    "difficulty": {
      "band": "upper_secondary",
      "level": 3,
      "skills": [
        "分塊聯集族的資格與最小閉包"
      ]
    },
    "variant": {
      "family_id": "qf-density13-event-partition-family-qualification",
      "seed": 202610136072,
      "parameters": {
        "n": 8,
        "blocks": [
          [
            0,
            3,
            4
          ],
          [
            1
          ],
          [
            2
          ],
          [
            5
          ],
          [
            6
          ],
          [
            7
          ]
        ],
        "E": [
          0
        ],
        "event_model": "partition_unions",
        "operation": "density13_event_partition_family_qualification",
        "domain": "finite_declared_event_family",
        "semantic_template": "qf-density13-event-partition-family-qualification/[false,[[0,3,4]],[0,3,4]]"
      }
    },
    "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": "2026-10-09",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261009-advance-20261013-r06-event-0072",
    "batch_id": "batch-20261009-advance-20261013-r06-event-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "b6f65c9045b98fa2cc7c00070086881cff5ba2c61f701308f6b83b87e6df6f7b",
      "family_id": "qf-density13-event-partition-family-qualification",
      "seed": 202610136072,
      "semantic_template": "qf-density13-event-partition-family-qualification/[false,[[0,3,4]],[0,3,4]]"
    },
    "fields": {
      "title": "Partition-Union Event Eligibility and Least Containing Event",
      "stem": "S=Fin(8)={0,…,7}; F consists of unions of complete disjoint nonempty blocks=[[0, 3, 4], [1], [2], [5], [6], [7]]. Candidate E=[0]⊆S. Give (is E a union of whole blocks?, partially hit block-member lists, smallest F set containing E). Sort inside each block and order blocks numerically lexicographically.",
      "answer": {
        "kind": "expression",
        "value": "(No, [[0, 3, 4]], [0, 3, 4])",
        "explanation": "Each block is taken wholly or not at all; the least containing event takes every block hit by E, while fully hit blocks are not partial hits. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is (No, [[0, 3, 4]], [0, 3, 4])."
      },
      "skills": [
        "Partition-Union Event Eligibility and Least Containing Event"
      ],
      "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": "S=Fin(n)，n=8；F為完整分塊的聯集族；完整互斥非空blocks=[[0, 3, 4], [1], [2], [5], [6], [7]]，候選E=[0]⊆S。依序答 (E是否完整分塊聯集, 部分命中的分塊成員列表, 最小包含E的F集合)，塊內排序、塊外按數值字典序。",
        "en": "S=Fin(8)={0,…,7}; F consists of unions of complete disjoint nonempty blocks=[[0, 3, 4], [1], [2], [5], [6], [7]]. Candidate E=[0]⊆S. Give (is E a union of whole blocks?, partially hit block-member lists, smallest F set containing E). Sort inside each block and order blocks numerically lexicographically."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "S=Fin(8)=[0, 1, 2, 3, 4, 5, 6, 7];declared event model=partition_unions",
      "after": "finite S alone does not specify its event family or a probability law",
      "relation": "implies",
      "rule_id": "declared_finite_event_family",
      "conditions": [],
      "explanation": {
        "zh-Hant": "有限樣本空間不自動指定F=P(S)；此處依題目宣告的事件族運算，集合資料不給概率、等可能性或獨立性。",
        "en": "A finite sample space does not automatically choose F=P(S).Use the declared event family;set data alone give no probabilities,equiprobability or independence."
      },
      "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": "ordered blocks=[[0, 3, 4], [1], [2], [5], [6], [7]];S=[0, 1, 2, 3, 4, 5, 6, 7]",
      "after": "nonempty,pairwise disjoint blocks cover S exactly",
      "relation": "implies",
      "rule_id": "complete_event_partition_premise",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每塊非空、塊間不交且覆蓋全部S；先核對完整分塊前提，不把任意子集合族當成分塊。",
        "en": "Verify nonempty disjoint blocks covering all S;an arbitrary family of subsets is not automatically a partition."
      },
      "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": "block=[0, 3, 4];E selects=[0]",
      "after": "hit=是;whole block selected=否",
      "relation": "implies",
      "rule_id": "partition_event_block_audit",
      "conditions": [],
      "explanation": {
        "zh-Hant": "F中的集合只能整塊取或整塊不取；部分命中仍是S的子集合，但不屬這個F。",
        "en": "A member of F must include or omit each entire block.A partial hit is still a subset of S but is not in this declared F."
      },
      "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": "block=[1];E selects=[]",
      "after": "hit=否;whole block selected=否",
      "relation": "implies",
      "rule_id": "partition_event_block_audit",
      "conditions": [],
      "explanation": {
        "zh-Hant": "F中的集合只能整塊取或整塊不取；部分命中仍是S的子集合，但不屬這個F。",
        "en": "A member of F must include or omit each entire block.A partial hit is still a subset of S but is not in this declared F."
      },
      "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": "block=[2];E selects=[]",
      "after": "hit=否;whole block selected=否",
      "relation": "implies",
      "rule_id": "partition_event_block_audit",
      "conditions": [],
      "explanation": {
        "zh-Hant": "F中的集合只能整塊取或整塊不取；部分命中仍是S的子集合，但不屬這個F。",
        "en": "A member of F must include or omit each entire block.A partial hit is still a subset of S but is not in this declared F."
      },
      "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": "block=[5];E selects=[]",
      "after": "hit=否;whole block selected=否",
      "relation": "implies",
      "rule_id": "partition_event_block_audit",
      "conditions": [],
      "explanation": {
        "zh-Hant": "F中的集合只能整塊取或整塊不取；部分命中仍是S的子集合，但不屬這個F。",
        "en": "A member of F must include or omit each entire block.A partial hit is still a subset of S but is not in this declared F."
      },
      "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": "block=[6];E selects=[]",
      "after": "hit=否;whole block selected=否",
      "relation": "implies",
      "rule_id": "partition_event_block_audit",
      "conditions": [],
      "explanation": {
        "zh-Hant": "F中的集合只能整塊取或整塊不取；部分命中仍是S的子集合，但不屬這個F。",
        "en": "A member of F must include or omit each entire block.A partial hit is still a subset of S but is not in this declared F."
      },
      "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": "block=[7];E selects=[]",
      "after": "hit=否;whole block selected=否",
      "relation": "implies",
      "rule_id": "partition_event_block_audit",
      "conditions": [],
      "explanation": {
        "zh-Hant": "F中的集合只能整塊取或整塊不取；部分命中仍是S的子集合，但不屬這個F。",
        "en": "A member of F must include or omit each entire block.A partial hit is still a subset of S but is not in this declared F."
      },
      "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": "partial blocks=[[0, 3, 4]];all hit blocks=[[0, 3, 4]]",
      "after": "qualified event=否;smallest containing F-set=[0, 3, 4]",
      "relation": "implies",
      "rule_id": "partition_union_minimal_event_hull",
      "conditions": [],
      "explanation": {
        "zh-Hant": "任何包含E的分塊聯集必包含全部命中塊，其聯集本身屬F，故這是最小包含集合；不是未宣告拓撲的閉包。",
        "en": "Any block-union containing E must include every hit block.Their union itself belongs to F,so it is the smallest containing F-set,not a closure in an unspecified topology."
      },
      "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,
      [
        [
          0,
          3,
          4
        ]
      ],
      [
        0,
        3,
        4
      ]
    ],
    "hit_blocks": [
      [
        0,
        3,
        4
      ]
    ]
  },
  "conclusion": {
    "canonical_value": "(否, [[0, 3, 4]], [0, 3, 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-20261009-advance-20261013-r06-event-foundations",
      "batch_path": "public/data/questions/batches/batch-20261009-advance-20261013-r06-event-foundations/questions.jsonl",
      "batch_sha256": "2ed3cc2f0adb9a12f8e2c4de29b320306a65fe7af8296a1f1c5d0b14bc7776d9",
      "original_line": 72,
      "original_line_sha256": "b68a86cfbe2841ffa641c21ed46f7d7ef76529c1637950cf14ab165186553c87",
      "question_compact_sha256": "b6f65c9045b98fa2cc7c00070086881cff5ba2c61f701308f6b83b87e6df6f7b",
      "english_row_compact_sha256": "dc9f8587868aaa751816b51f7b0ea3e4b02519d92e518fc281a77dc2aaed1e4d",
      "english_manifest_path": "public/data/i18n/editions/batch-20261009-advance-20261013-r06-event-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261009-advance-20261013-r06-event-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-event": {
        "sha256": "5962ae6672cb79c34888bdda4ac66de557a4f9b21243f2445414d7dd326b9c74",
        "english_sha256": "12d8199c90226e08b52a3f9cf27ce44053dc8473e8bab4280145aea146a4c261",
        "original_path": "public/data/mko/event.json",
        "english_path": "public/data/i18n/editions/mko-event.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "5b2ce631281eb24b279908f47c6de66fa60d326520852879d324c4f3ac653970",
    "original_candidate_sha256": "8af681924ca1be4080a1841bec3359967f223265f0ab7cc6e0b7e320e0a6f15b",
    "original_candidate_line_sha256": "8af681924ca1be4080a1841bec3359967f223265f0ab7cc6e0b7e320e0a6f15b"
  },
  "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-p12",
    "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": "b9bbc1c947f477a9c4bb5a579368ac5265f2da4572ce612c30f3633ecaf4ae80",
      "question_first_witness_sha256": "93f720df3ab3af13a792a3e6b443728e3c2a56276df839911f36569333236c55"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 9,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "c9a67de4bc0c3601cba3cbdfbf34bdf9abac7ae9adcb67d6e7e57a9a1fcd737e",
      "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."
  }
}