{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261008-advance-20261011-r05-subset-0017",
  "pilot_number": "R0009-05167",
  "number": 5167,
  "family_id": "qf-density11-subset-obstruction-list",
  "title": {
    "zh-Hant": "第 5167 題：非嚴格包含的全部阻擋元素",
    "en": "Question 5167: Complete Inclusion Obstruction Set"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261008-advance-20261011-r05-subset-0017",
    "batch_id": "batch-20261008-advance-20261011-r05-subset-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-subset"
    ],
    "title_zh": "非嚴格包含的全部阻擋元素",
    "stem_zh": "有限整數集合；同一整數型別；A=[-16, 16, 0]、B=[0]。完整列A⊆B的阻擋元素，即所有屬A但不屬B者，遞增且去重。",
    "answer": {
      "kind": "expression",
      "value": "[-16, 16]",
      "explanation_zh": "全部阻擋元素=[-16, 16]。阻擋者正是A\\B；沒有阻擋才通過包含，空A不產生反例。"
    },
    "difficulty": {
      "band": "lower_secondary",
      "level": 2,
      "skills": [
        "非嚴格包含的全部阻擋元素"
      ]
    },
    "variant": {
      "family_id": "qf-density11-subset-obstruction-list",
      "seed": 202610115017,
      "parameters": {
        "A": [
          -16,
          16,
          0
        ],
        "B": [
          0
        ],
        "operation": "density11_subset_obstruction_list",
        "domain": "finite_integer_set_slice",
        "semantic_template": "qf-density11-subset-obstruction-list/[-16,16]"
      }
    },
    "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-r05-subset-0017",
    "batch_id": "batch-20261008-advance-20261011-r05-subset-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "aa1bc61b1bf5157573e8fa996ca16a1c3b3fa8d7d02111a7c532faea1599e0ed",
      "family_id": "qf-density11-subset-obstruction-list",
      "seed": 202610115017,
      "semantic_template": "qf-density11-subset-obstruction-list/[-16,16]"
    },
    "fields": {
      "title": "Complete Inclusion Obstruction Set",
      "stem": "Finite integer sets A=[-16, 16, 0] and B=[0]; repeated elements count once. List all obstructions to A⊆B, namely elements in A but not B, increasingly without repetition.",
      "answer": {
        "kind": "expression",
        "value": "[-16, 16]",
        "explanation": "The obstruction set is A\\B; inclusion holds exactly when it is empty, and an empty A supplies no counterexample. The exact result, retaining the requested order, is [-16, 16]."
      },
      "skills": [
        "Complete Inclusion Obstruction Set"
      ],
      "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": "有限整數集合；同一整數型別；A=[-16, 16, 0]、B=[0]。完整列A⊆B的阻擋元素，即所有屬A但不屬B者，遞增且去重。",
        "en": "Finite integer sets A=[-16, 16, 0] and B=[0]; repeated elements count once. List all obstructions to A⊆B, namely elements in A but not B, increasingly without repetition."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "A=[-16, 0, 16];B=[0];same integer type",
      "after": "A⊆B means every A-element belongs to B",
      "relation": "implies",
      "rule_id": "set_inclusion_definition",
      "conditions": [],
      "explanation": {
        "zh-Hant": "包含的反例必須屬A且不屬B；域外A的數字不是見證，空A不產生阻擋。",
        "en": "An inclusion obstruction must belong to A but not B;an element outside A is no witness,and empty A gives no obstruction."
      },
      "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": "A\\B:source element=-16;target=[0]",
      "after": "obstruction=是",
      "relation": "implies",
      "rule_id": "inclusion_obstruction_row",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐一核對來源集合的全部元素，目標集合不含才列為阻擋。",
        "en": "Audit every source-set element;list it as an obstruction exactly when the target omits it."
      },
      "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": "A\\B:source element=0;target=[0]",
      "after": "obstruction=否",
      "relation": "implies",
      "rule_id": "inclusion_obstruction_row",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐一核對來源集合的全部元素，目標集合不含才列為阻擋。",
        "en": "Audit every source-set element;list it as an obstruction exactly when the target omits it."
      },
      "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": "A\\B:source element=16;target=[0]",
      "after": "obstruction=是",
      "relation": "implies",
      "rule_id": "inclusion_obstruction_row",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐一核對來源集合的全部元素，目標集合不含才列為阻擋。",
        "en": "Audit every source-set element;list it as an obstruction exactly when the target omits it."
      },
      "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": "[-16, 16]",
      "after": "A⊆B=否",
      "relation": "implies",
      "rule_id": "complete_inclusion_obstruction_inventory",
      "conditions": [],
      "explanation": {
        "zh-Hant": "因已核對全部A，阻擋集空當且僅當這個有限包含成立。",
        "en": "Because every A-element was checked,the empty obstruction set is equivalent to this finite inclusion."
      },
      "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": [
      -16,
      16
    ]
  },
  "conclusion": {
    "canonical_value": "[-16, 16]",
    "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-r05-subset-foundations",
      "batch_path": "public/data/questions/batches/batch-20261008-advance-20261011-r05-subset-foundations/questions.jsonl",
      "batch_sha256": "faef8ba0762c4b1138b5f3bb62aed94757b8580fef3c3b9df5518b48ec561db7",
      "original_line": 17,
      "original_line_sha256": "bad390067023d4358e0a5fff90baa61bf29958f906106b488d65b62210491526",
      "question_compact_sha256": "aa1bc61b1bf5157573e8fa996ca16a1c3b3fa8d7d02111a7c532faea1599e0ed",
      "english_row_compact_sha256": "c79ef9cb201030bf7750aefd6806d17442e90a4601b6af368ec6affa02ea0fb7",
      "english_manifest_path": "public/data/i18n/editions/batch-20261008-advance-20261011-r05-subset-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261008-advance-20261011-r05-subset-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-subset": {
        "sha256": "984a77e67dc9322e36b869843513f1cb76f9527912202782848120c123ce99b3",
        "english_sha256": "af2446cfbbb6eff0aba71545b87591295e79df30d0992a5085cf44be4170d534",
        "original_path": "public/data/mko/subset.json",
        "english_path": "public/data/i18n/editions/mko-subset.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "3132b1200be7a5de38191badc52fb1727d4dd9d4c2f26818de2e8a76426d2d3f",
    "original_candidate_sha256": "b9dcf7b296073b33bfe7b1d1df5ddeab348581ecf5e8533ad4df39d5ac0e3828",
    "original_candidate_line_sha256": "b9dcf7b296073b33bfe7b1d1df5ddeab348581ecf5e8533ad4df39d5ac0e3828"
  },
  "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": 5,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "b8ec94376844fbaa6bfcd024e112fb68f466e87de8211864073d89ee597cfa9c",
      "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."
  }
}