{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261008-advance-20261010-r07-quantifier-0065",
  "pilot_number": "R0007-04315",
  "number": 4315,
  "family_id": "qf-density10-quantifier-candidate-mask",
  "title": {
    "zh-Hant": "第 4315 題：三個候選的域內反例資格",
    "en": "Question 4315: In-Domain Counterexample Candidate Mask"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261008-advance-20261010-r07-quantifier-0065",
    "batch_id": "batch-20261008-advance-20261010-r07-quantifier-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-quantifier"
    ],
    "title_zh": "三個候選的域內反例資格",
    "stem_zh": "有限明示論域 U=[-3, -2, -1, 1, 2, 3]；可判定述詞P(x)為x≥0。有序候選=[1, -2, 3]。逐項答是否同時屬U且不滿足P，用是或否；域外不合資格。",
    "answer": {
      "kind": "expression",
      "value": "[否, 是, 否]",
      "explanation_zh": "域內反例資格=[否, 是, 否]。先查論域，再查述詞否定；域外、域內滿足者都不是反例，不外推無限論域。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 2,
      "skills": [
        "三個候選的域內反例資格"
      ]
    },
    "variant": {
      "family_id": "qf-density10-quantifier-candidate-mask",
      "seed": 202610107065,
      "parameters": {
        "U": [
          -3,
          -2,
          -1,
          1,
          2,
          3
        ],
        "predicate": "ge",
        "parameter": 0,
        "candidates": [
          1,
          -2,
          3
        ],
        "operation": "density10_quantifier_candidate_mask",
        "domain": "finite_integer_domain",
        "semantic_template": "qf-density10-quantifier-candidate-mask/[\"satisfies\",\"violates\",\"satisfies\"]"
      }
    },
    "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-0065",
    "batch_id": "batch-20261008-advance-20261010-r07-quantifier-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "6470fdfcd5852aeafc7c2a13050a627dec98477a0ede44a56a911c526096bb91",
      "family_id": "qf-density10-quantifier-candidate-mask",
      "seed": 202610107065,
      "semantic_template": "qf-density10-quantifier-candidate-mask/[\"satisfies\",\"violates\",\"satisfies\"]"
    },
    "fields": {
      "title": "In-Domain Counterexample Candidate Mask",
      "stem": "The explicitly finite integer domain is U=[-3, -2, -1, 1, 2, 3] and P(x) means x≥0. Ordered candidates=[1, -2, 3]. For each, answer Yes/No to membership in U together with failure of P; outside candidates are ineligible.",
      "answer": {
        "kind": "expression",
        "value": "[No, Yes, No]",
        "explanation": "Check the domain before the negated predicate; outside candidates and satisfying domain elements are not counterexamples. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is [No, Yes, No]."
      },
      "skills": [
        "In-Domain Counterexample Candidate Mask"
      ],
      "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=[-3, -2, -1, 1, 2, 3]；可判定述詞P(x)為x≥0。有序候選=[1, -2, 3]。逐項答是否同時屬U且不滿足P，用是或否；域外不合資格。",
        "en": "The explicitly finite integer domain is U=[-3, -2, -1, 1, 2, 3] and P(x) means x≥0. Ordered candidates=[1, -2, 3]. For each, answer Yes/No to membership in U together with failure of P; outside candidates are ineligible."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "complete finite U=[-3, -2, -1, 1, 2, 3]",
      "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=-3;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=-2;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=-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": "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=2;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": "x=3;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": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "[[-3, 否], [-2, 否], [-1, 否], [1, 是], [2, 是], [3, 是]]",
      "after": "T=[1, 2, 3];F=[-3, -2, -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
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "candidate=1;inside=是;predicate if inside=是",
      "after": "否",
      "relation": "implies",
      "rule_id": "ordered_counterexample_candidate",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先檢查在域內，再查述詞失敗；候選列保持原順序及重複位置，域外不具反例資格。",
        "en": "Check membership before predicate failure;preserve candidate order and repeated positions,and reject outside-domain 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": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "candidate=-2;inside=是;predicate if inside=否",
      "after": "是",
      "relation": "implies",
      "rule_id": "ordered_counterexample_candidate",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先檢查在域內，再查述詞失敗；候選列保持原順序及重複位置，域外不具反例資格。",
        "en": "Check membership before predicate failure;preserve candidate order and repeated positions,and reject outside-domain 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": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "candidate=3;inside=是;predicate if inside=是",
      "after": "否",
      "relation": "implies",
      "rule_id": "ordered_counterexample_candidate",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先檢查在域內，再查述詞失敗；候選列保持原順序及重複位置，域外不具反例資格。",
        "en": "Check membership before predicate failure;preserve candidate order and repeated positions,and reject outside-domain 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
    }
  ],
  "math_model": {
    "result": [
      false,
      true,
      false
    ],
    "true_elements": [
      1,
      2,
      3
    ],
    "false_elements": [
      -3,
      -2,
      -1
    ],
    "candidate_rows": [
      [
        1,
        true,
        true,
        false
      ],
      [
        -2,
        true,
        false,
        true
      ],
      [
        3,
        true,
        true,
        false
      ]
    ]
  },
  "conclusion": {
    "canonical_value": "[否, 是, 否]",
    "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": 65,
      "original_line_sha256": "2f50920abac9d7a0134691b8d35338953e9938670aad4efbdaa1c5d9317601ea",
      "question_compact_sha256": "6470fdfcd5852aeafc7c2a13050a627dec98477a0ede44a56a911c526096bb91",
      "english_row_compact_sha256": "11982b6c2d2f07a220cd53697bde20bf8f9f8474a04df6361a9ff3fc2f3010b6",
      "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": "a3482fc6f2f17d2f99382adcf1264c24df11e622b0fa11b4fc4921b11969d9d0",
    "original_candidate_line_sha256": "a3482fc6f2f17d2f99382adcf1264c24df11e622b0fa11b4fc4921b11969d9d0"
  },
  "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": 11,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "ca2da0347ee0c85194d663f5ee0315f3268686b805bcccd45508ae33d0d34f6f",
      "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."
  }
}