{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261013-advance-20261019-r02-neighborhood-0011",
  "pilot_number": "R0018-09786",
  "number": 9786,
  "family_id": "qf-ext19-neighborhood-witness",
  "title": {
    "zh-Hant": "第 9786 題：鄰域與開集見證",
    "en": "Question 9786: Point Guard and Every Open Neighborhood Witness"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261013-advance-20261019-r02-neighborhood-0011",
    "batch_id": "batch-20261013-advance-20261019-r02-neighborhood-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-neighborhood"
    ],
    "title_zh": "鄰域與開集見證",
    "stem_zh": "完整有限拓撲與指定點。操作=neighborhood-witness；資料={\"n\":4,\"opens\":[[],[2],[2,3],[0,2,3],[0,1,2,3]],\"x\":2,\"candidate\":[0,2,3]}。請依 [n,x,point_qualified,candidate,open_witnesses_or_不求值,neighborhood_or_不求值] 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(4, 2, 是, [0, 2, 3], [[2], [2, 3], [0, 2, 3]], 是)",
      "explanation_zh": "第二批擴展核對=(4, 2, 是, [0, 2, 3], [[2], [2, 3], [0, 2, 3]], 是)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；鄰域包含含點開集；鄰域本身不必開；域外點不求值。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 2,
      "skills": [
        "鄰域與開集見證"
      ]
    },
    "variant": {
      "family_id": "qf-ext19-neighborhood-witness",
      "seed": 202610192011,
      "parameters": {
        "n": 4,
        "opens": [
          [],
          [
            2
          ],
          [
            2,
            3
          ],
          [
            0,
            2,
            3
          ],
          [
            0,
            1,
            2,
            3
          ]
        ],
        "x": 2,
        "candidate": [
          0,
          2,
          3
        ],
        "domain": "Fin_topological_neighborhood",
        "operation": "neighborhood-witness",
        "semantic_template": "qf-ext19-neighborhood-witness/{\"candidate\":[0,2,3],\"n\":4,\"opens\":[[],[2],[2,3],[0,2,3],[0,1,2,3]],\"x\":2}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_finite_model",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、Lean執行或形式Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-13",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261013-advance-20261019-r02-neighborhood-0011",
    "batch_id": "batch-20261013-advance-20261019-r02-neighborhood-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "f7b1fd73730c138f18b18ba36bd9909e42da8f7934c816aa6e1fe72404120e9c",
      "family_id": "qf-ext19-neighborhood-witness",
      "seed": 202610192011,
      "semantic_template": "qf-ext19-neighborhood-witness/{\"candidate\":[0,2,3],\"n\":4,\"opens\":[[],[2],[2,3],[0,2,3],[0,1,2,3]],\"x\":2}"
    },
    "fields": {
      "title": "Point Guard and Every Open Neighborhood Witness",
      "stem": "Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [2], [2, 3], [0, 2, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Point x=2, candidate N=[0, 2, 3]. Give (n, x, point in X?, N, all open U with x∈U⊆N or not-evaluated, neighborhood? or not-evaluated). N itself need not be open.",
      "answer": {
        "kind": "expression",
        "value": "(4, 2, Yes, [0, 2, 3], [[2], [2, 3], [0, 2, 3]], Yes)",
        "explanation": "Type-check the point before evaluating neighborhood claims, then enumerate every open witness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 2, Yes, [0, 2, 3], [[2], [2, 3], [0, 2, 3]], Yes)."
      },
      "skills": [
        "Point Guard and Every Open Neighborhood Witness"
      ],
      "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": "完整有限拓撲與指定點。操作=neighborhood-witness；資料={\"n\":4,\"opens\":[[],[2],[2,3],[0,2,3],[0,1,2,3]],\"x\":2,\"candidate\":[0,2,3]}。請依 [n,x,point_qualified,candidate,open_witnesses_or_不求值,neighborhood_or_不求值] 欄序回答。",
        "en": "Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [2], [2, 3], [0, 2, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Point x=2, candidate N=[0, 2, 3]. Give (n, x, point in X?, N, all open U with x∈U⊆N or not-evaluated, neighborhood? or not-evaluated). N itself need not be open."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "n=4;raw open family=[[], [2], [2, 3], [0, 2, 3], [0, 1, 2, 3]]",
      "after": "universe=[0, 1, 2, 3];normalized opens=[[], [2], [2, 3], [0, 2, 3], [0, 1, 2, 3]];empty=是;full=是;union closed=是;intersection closed=是",
      "relation": "implies",
      "rule_id": "given_complete_finite_topology_guard",
      "conditions": [],
      "explanation": {
        "zh-Hant": "子集與族去重後按mask排序，完整核空集、全集及二元聯交閉合。有限底集合使任意聯集只需有限不同成員；本題要求真拓撲，非法資料不拿來算後續性質。",
        "en": "Deduplicate subsets/families and order by masks.Check empty/whole membership and all binary union/intersection closure.On a finite underlying set,arbitrary unions reduce to finitely many distinct members.These tasks require an actual topology;invalid data is not used for later properties."
      },
      "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": "point candidate=2;underlying Fin(4)",
      "after": "point qualified=是",
      "relation": "implies",
      "rule_id": "neighborhood_point_domain_guard",
      "conditions": [],
      "explanation": {
        "zh-Hant": "域外點不作鄰域判斷，不由負索引或空見證表製造假；候選集合的型別仍須獨立合格。",
        "en": "An outside point leaves neighborhood claims unevaluated,not False via negative indexing or an empty witness list.Candidate sets must still independently be typed."
      },
      "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": "raw candidate=[0, 2, 3]",
      "after": "typed normalized candidate=[0, 2, 3]",
      "relation": "implies",
      "rule_id": "neighborhood_candidate_set_normalization",
      "conditions": [],
      "explanation": {
        "zh-Hant": "集合去重與點資格是分開的資料條件。",
        "en": "Set normalization and point qualification are separate data conditions."
      },
      "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": "point=2;complete open family=[[], [2], [2, 3], [0, 2, 3], [0, 1, 2, 3]]",
      "after": "all point-containing opens=[[2], [2, 3], [0, 2, 3], [0, 1, 2, 3]]",
      "relation": "implies",
      "rule_id": "complete_point_open_witness_basis",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只取含該合法點的所有開集，作完整見證候選；鄰域本身不必開，只需包含其中至少一個。",
        "en": "Take every open containing the qualified point as the complete witness candidates.A neighborhood itself need not be open;it must contain at least one of them."
      },
      "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": "phase=candidate;point=2;candidate=[0, 2, 3];point-open basis=[[2], [2, 3], [0, 2, 3], [0, 1, 2, 3]]",
      "after": "all contained open witnesses=[[2], [2, 3], [0, 2, 3]];neighborhood=是",
      "relation": "implies",
      "rule_id": "all_open_neighborhood_witnesses",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依x∈U⊆candidate列出所有開見證；有一個即是鄰域，沒有才否定，不能只看candidate是否開或是否含點。",
        "en": "List all open witnesses satisfying x∈U⊆candidate.One witness suffices;absence refutes the property.Do not decide only by the candidate's openness or point membership."
      },
      "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": [
      4,
      2,
      true,
      [
        0,
        2,
        3
      ],
      [
        [
          2
        ],
        [
          2,
          3
        ],
        [
          0,
          2,
          3
        ]
      ],
      true
    ]
  },
  "conclusion": {
    "canonical_value": "(4, 2, 是, [0, 2, 3], [[2], [2, 3], [0, 2, 3]], 是)",
    "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-20261013-advance-20261019-r02-neighborhood-foundations",
      "batch_path": "public/data/questions/batches/batch-20261013-advance-20261019-r02-neighborhood-foundations/questions.jsonl",
      "batch_sha256": "8c12ac9970811934974e25ebbfac4514b8e30b9256f6ced5cd470d4dd9c23482",
      "original_line": 11,
      "original_line_sha256": "c5fe47c30b955a64eab33086dc7f75043816737ba8da0bf0a08496fb9d8eb4e7",
      "question_compact_sha256": "f7b1fd73730c138f18b18ba36bd9909e42da8f7934c816aa6e1fe72404120e9c",
      "english_row_compact_sha256": "c98414195e071376f6fa3167ce127fbd9bdae8fdd0a16650a75eeb4a8123797b",
      "english_manifest_path": "public/data/i18n/editions/batch-20261013-advance-20261019-r02-neighborhood-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261013-advance-20261019-r02-neighborhood-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-neighborhood": {
        "sha256": "175c50e3f4cbb3dfa266c387a13492faeffdacb3df5d699b2ad94f7ea5f91ef1",
        "english_sha256": "1822e364cadbcc3eec3cbc1611d4b6cbc00624e445f49f039db873791a78d69d",
        "original_path": "public/data/mko/neighborhood.json",
        "english_path": "public/data/i18n/editions/mko-neighborhood.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "2cf5df7bf9c3d48dfd41d968723b60d910c4ecfec9c2089d22ac1cda059a6086",
    "original_candidate_sha256": "ea7c71d07fd091510c56859cc1a741a2be58f9a49e08a33c31e2b53e5b829a7c",
    "original_candidate_line_sha256": "ea7c71d07fd091510c56859cc1a741a2be58f9a49e08a33c31e2b53e5b829a7c"
  },
  "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-20261026-006-p19",
    "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": "bd35da62087a15d6b89b45585dc46e89b9efbe097cba7b3c707b5030496f013c",
      "question_first_witness_sha256": "fab45a10d8ac95c7d61d32e47d1bdd3ca862c17c3289a6bf4b01d4dc9492fbb6"
    },
    "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": "43f593303a2e5fd2f08e546b0b9519783340fca5980b1aab3fa1ad5cbe07946f",
      "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."
  }
}