{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261008-advance-20261011-r04-set-0066",
  "pilot_number": "R0009-05016",
  "number": 5016,
  "family_id": "qf-density11-set-candidate-partition",
  "title": {
    "zh-Hant": "第 5016 題：候選元素的隸屬分割",
    "en": "Question 5016: Partition of an Explicit Candidate Set"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261008-advance-20261011-r04-set-0066",
    "batch_id": "batch-20261008-advance-20261011-r04-set-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-set"
    ],
    "title_zh": "候選元素的隸屬分割",
    "stem_zh": "有限整數集合；A=[-15, 0]、候選原列=[-15, 15, 0, 15]。候選視作集合，依序列 (候選中屬A者, 候選中不屬A者)，各列遞增。",
    "answer": {
      "kind": "expression",
      "value": "([-15, 0], [15])",
      "explanation_zh": "候選元素分割=([-15, 0], [15])。先去候選重複再判隸屬；A中未被列為候選的額外元素不進輸出。"
    },
    "difficulty": {
      "band": "lower_secondary",
      "level": 2,
      "skills": [
        "候選元素的隸屬分割"
      ]
    },
    "variant": {
      "family_id": "qf-density11-set-candidate-partition",
      "seed": 202610114066,
      "parameters": {
        "A": [
          -15,
          0
        ],
        "candidates": [
          -15,
          15,
          0,
          15
        ],
        "operation": "density11_set_candidate_partition",
        "domain": "finite_integer_set_slice",
        "semantic_template": "qf-density11-set-candidate-partition/[[-15,0],[15]]"
      }
    },
    "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-r04-set-0066",
    "batch_id": "batch-20261008-advance-20261011-r04-set-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "0a7258fc177ea1cd091fb6e40d78ac003147fb7a198d81ca11b487b3f1716e9a",
      "family_id": "qf-density11-set-candidate-partition",
      "seed": 202610114066,
      "semantic_template": "qf-density11-set-candidate-partition/[[-15,0],[15]]"
    },
    "fields": {
      "title": "Partition of an Explicit Candidate Set",
      "stem": "Finite integer set A=[-15, 0]; raw candidates=[-15, 15, 0, 15] are treated as a set. Give (candidates belonging to A, candidates outside A), each sorted increasingly.",
      "answer": {
        "kind": "expression",
        "value": "([-15, 0], [15])",
        "explanation": "Normalize candidate repetition before membership tests; unlisted extra elements of A do not enter either output part. The exact result, retaining the requested order, is ([-15, 0], [15])."
      },
      "skills": [
        "Partition of an Explicit Candidate 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；A=[-15, 0]、候選原列=[-15, 15, 0, 15]。候選視作集合，依序列 (候選中屬A者, 候選中不屬A者)，各列遞增。",
        "en": "Finite integer set A=[-15, 0]; raw candidates=[-15, 15, 0, 15] are treated as a set. Give (candidates belonging to A, candidates outside A), each sorted increasingly."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "raw integer list=[-15, 0]",
      "after": "extensional set=[-15, 0]",
      "relation": "implies",
      "rule_id": "set_extensionality_normalization",
      "conditions": [],
      "explanation": {
        "zh-Hant": "集合只看有哪些元素；去重及數值排序不改隸屬，並保留原始清單作來源。",
        "en": "A set depends on its elements;deduplication and numeric ordering do not change membership.The original list is retained as source."
      },
      "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": "candidate=-15;A=[-15, 0]",
      "after": "in A=是",
      "relation": "implies",
      "rule_id": "candidate_set_partition_row",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只分割候選集合C，不把A中未列成候選的額外元素加進輸出。",
        "en": "Partition only candidate set C;do not add A-elements that were not candidates."
      },
      "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": "candidate=0;A=[-15, 0]",
      "after": "in A=是",
      "relation": "implies",
      "rule_id": "candidate_set_partition_row",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只分割候選集合C，不把A中未列成候選的額外元素加進輸出。",
        "en": "Partition only candidate set C;do not add A-elements that were not candidates."
      },
      "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": "candidate=15;A=[-15, 0]",
      "after": "in A=否",
      "relation": "implies",
      "rule_id": "candidate_set_partition_row",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只分割候選集合C，不把A中未列成候選的額外元素加進輸出。",
        "en": "Partition only candidate set C;do not add A-elements that were not candidates."
      },
      "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": "[-15, 0, 15]",
      "after": "([-15, 0], [15])",
      "relation": "implies",
      "rule_id": "candidate_set_partition_complete",
      "conditions": [],
      "explanation": {
        "zh-Hant": "兩段互不交且合起來恰是去重候選集，先列屬A者再列不屬者。",
        "en": "The two parts are disjoint and cover the deduplicated candidate set;list members of A before nonmembers."
      },
      "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": [
      [
        -15,
        0
      ],
      [
        15
      ]
    ]
  },
  "conclusion": {
    "canonical_value": "([-15, 0], [15])",
    "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-r04-set-foundations",
      "batch_path": "public/data/questions/batches/batch-20261008-advance-20261011-r04-set-foundations/questions.jsonl",
      "batch_sha256": "f470bac7d0a7bba63071eb9254603f66dd92d4c55f5ceb13a0d1f7850edeb98e",
      "original_line": 66,
      "original_line_sha256": "2bc374fd6b3089224d421ce0bcee6f9f59d4d0f33599f30e148b3fafa1f85f80",
      "question_compact_sha256": "0a7258fc177ea1cd091fb6e40d78ac003147fb7a198d81ca11b487b3f1716e9a",
      "english_row_compact_sha256": "816810dce57a4c46d23aca914d358637b8b49841e2e2fbd112d63f201a1202ea",
      "english_manifest_path": "public/data/i18n/editions/batch-20261008-advance-20261011-r04-set-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261008-advance-20261011-r04-set-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-set": {
        "sha256": "4f736510823c29158d78264f57fa9f96c5c2107ddc99999622656fdacc7a5728",
        "english_sha256": "f1024a1b7bcb63048a9ba5c6ca166a8236e5d37f63c4c64d3985f769c7ae742c",
        "original_path": "public/data/mko/set.json",
        "english_path": "public/data/i18n/editions/mko-set.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "3132b1200be7a5de38191badc52fb1727d4dd9d4c2f26818de2e8a76426d2d3f",
    "original_candidate_sha256": "e05920a0b6a60a675d9b169b63860a66fc24c4f7e97a61e2165b71c2251de372",
    "original_candidate_line_sha256": "e05920a0b6a60a675d9b169b63860a66fc24c4f7e97a61e2165b71c2251de372"
  },
  "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": "20dae5a7cdb32722b955d411926f6acac5caee8569b739094445065a0f99e16a",
      "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."
  }
}