{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261008-advance-20261010-r07-quantifier-0025",
  "pilot_number": "R0007-04400",
  "number": 4400,
  "family_id": "qf-density10-quantifier-truth-partition",
  "title": {
    "zh-Hant": "第 4400 題：有限域的真假元素分割",
    "en": "Question 4400: Complete Finite Predicate Partition"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261008-advance-20261010-r07-quantifier-0025",
    "batch_id": "batch-20261008-advance-20261010-r07-quantifier-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-quantifier"
    ],
    "title_zh": "有限域的真假元素分割",
    "stem_zh": "有限明示論域 U=[-24, -1, 0, 1, 24]；可判定述詞P(x)為x≥0。依序列 (滿足P的元素, 不滿足P的元素)，兩列整數遞增。",
    "answer": {
      "kind": "expression",
      "value": "([0, 1, 24], [-24, -1])",
      "explanation_zh": "真與假元素分割=([0, 1, 24], [-24, -1])。完整逐元素測述詞，真假兩段覆蓋U且不重疊；不外推無限論域。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 3,
      "skills": [
        "有限域的真假元素分割"
      ]
    },
    "variant": {
      "family_id": "qf-density10-quantifier-truth-partition",
      "seed": 202610107025,
      "parameters": {
        "U": [
          -24,
          -1,
          0,
          1,
          24
        ],
        "predicate": "ge",
        "parameter": 0,
        "operation": "density10_quantifier_truth_partition",
        "domain": "finite_integer_domain",
        "semantic_template": "qf-density10-quantifier-truth-partition/[[0,1,24],[-24,-1]]"
      }
    },
    "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-20261010-r07-quantifier-0025",
    "batch_id": "batch-20261008-advance-20261010-r07-quantifier-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "e4a8ddf9b1eb58e70bfef24c5c0813c1654cbae06dca1806249d45c410852029",
      "family_id": "qf-density10-quantifier-truth-partition",
      "seed": 202610107025,
      "semantic_template": "qf-density10-quantifier-truth-partition/[[0,1,24],[-24,-1]]"
    },
    "fields": {
      "title": "Complete Finite Predicate Partition",
      "stem": "The explicitly finite integer domain is U=[-24, -1, 0, 1, 24] and P(x) means x≥0. Give (elements satisfying P, elements not satisfying P), each sorted increasingly.",
      "answer": {
        "kind": "expression",
        "value": "([0, 1, 24], [-24, -1])",
        "explanation": "Evaluate every domain element; the two disjoint parts cover U. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is ([0, 1, 24], [-24, -1])."
      },
      "skills": [
        "Complete Finite Predicate Partition"
      ],
      "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": "有限明示論域 U=[-24, -1, 0, 1, 24]；可判定述詞P(x)為x≥0。依序列 (滿足P的元素, 不滿足P的元素)，兩列整數遞增。",
        "en": "The explicitly finite integer domain is U=[-24, -1, 0, 1, 24] and P(x) means x≥0. Give (elements satisfying P, elements not satisfying P), each sorted increasingly."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "complete finite U=[-24, -1, 0, 1, 24]",
      "after": "universal on empty domain=True;existential on empty domain=False",
      "relation": "implies",
      "rule_id": "finite_quantifier_scope",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只枚舉明示有限論域，空域全稱真、存在假；完成有限枚舉不外推無限論域。",
        "en": "Enumerate only the declared finite domain;empty-domain universals are true and existentials false.Complete finite enumeration does not extend to an infinite domain."
      },
      "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": "x=-24;predicate=ge 0",
      "after": "P(x)=否",
      "relation": "implies",
      "rule_id": "finite_predicate_row",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐元素計算宣告的可判定整數述詞，真與假元素完整保留。",
        "en": "Evaluate the declared decidable integer predicate for each element,retaining both true and false cases."
      },
      "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": "x=-1;predicate=ge 0",
      "after": "P(x)=否",
      "relation": "implies",
      "rule_id": "finite_predicate_row",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐元素計算宣告的可判定整數述詞，真與假元素完整保留。",
        "en": "Evaluate the declared decidable integer predicate for each element,retaining both true and false cases."
      },
      "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": "x=0;predicate=ge 0",
      "after": "P(x)=是",
      "relation": "implies",
      "rule_id": "finite_predicate_row",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐元素計算宣告的可判定整數述詞，真與假元素完整保留。",
        "en": "Evaluate the declared decidable integer predicate for each element,retaining both true and false cases."
      },
      "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": "x=1;predicate=ge 0",
      "after": "P(x)=是",
      "relation": "implies",
      "rule_id": "finite_predicate_row",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐元素計算宣告的可判定整數述詞，真與假元素完整保留。",
        "en": "Evaluate the declared decidable integer predicate for each element,retaining both true and false cases."
      },
      "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": "x=24;predicate=ge 0",
      "after": "P(x)=是",
      "relation": "implies",
      "rule_id": "finite_predicate_row",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐元素計算宣告的可判定整數述詞，真與假元素完整保留。",
        "en": "Evaluate the declared decidable integer predicate for each element,retaining both true and false cases."
      },
      "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": "[[-24, 否], [-1, 否], [0, 是], [1, 是], [24, 是]]",
      "after": "T=[0, 1, 24];F=[-24, -1];T∪F=U and T∩F=∅",
      "relation": "implies",
      "rule_id": "complete_truth_partition",
      "conditions": [],
      "explanation": {
        "zh-Hant": "真假兩段互斥且覆蓋去重論域，沒有把未列的整數當作反例。",
        "en": "The true and false parts are disjoint and cover the deduplicated domain;unlisted integers are not counterexamples."
      },
      "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": [
      [
        0,
        1,
        24
      ],
      [
        -24,
        -1
      ]
    ]
  },
  "conclusion": {
    "canonical_value": "([0, 1, 24], [-24, -1])",
    "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-20261010-r07-quantifier-foundations",
      "batch_path": "public/data/questions/batches/batch-20261008-advance-20261010-r07-quantifier-foundations/questions.jsonl",
      "batch_sha256": "13b0fa8d8039c9cb766931f8811a087808fd7708cee37eff0ae37506a6f390f1",
      "original_line": 25,
      "original_line_sha256": "5ee30b00dd73462f3f095983419a2dd7b6d1de0f5a249a4297da9d9e84249518",
      "question_compact_sha256": "e4a8ddf9b1eb58e70bfef24c5c0813c1654cbae06dca1806249d45c410852029",
      "english_row_compact_sha256": "adfec881c6e06699f4037966959184ac05a59428bd9c5a5e18633bb1cb21c684",
      "english_manifest_path": "public/data/i18n/editions/batch-20261008-advance-20261010-r07-quantifier-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261008-advance-20261010-r07-quantifier-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-quantifier": {
        "sha256": "3b2f73bd57aa43b5441edef2b27d44470f8cbf04a229fb31e98243708e818341",
        "english_sha256": "cf34ab77acaa732f1cc6926b00bee27e73188f1108aaad401ca0a12fc391cc57",
        "original_path": "public/data/mko/quantifier.json",
        "english_path": "public/data/i18n/editions/mko-quantifier.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "de1fb611cf230f243ddd3334fbc53c9205cbe528b78346a10544972dede78b95",
    "original_candidate_sha256": "b7ee16669a991687807f548c557a03c3fda67434f361ae79bb21fc7c246d57f1",
    "original_candidate_line_sha256": "b7ee16669a991687807f548c557a03c3fda67434f361ae79bb21fc7c246d57f1"
  },
  "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-20261026-004-p8",
    "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": "faf0582c0184613e6fae207374056133eae600883a29c67d8438113bbe3c485a",
      "question_first_witness_sha256": "4b283b6ff424b1d22caa36f3552b0fa21b1b66a0a883d7cbe9cedc8dd77b2e42"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 7,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "5e254070307d04f88dc3032b63f59e9b193c042b5d8d1a083d313901568047b4",
      "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."
  }
}