{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261003-advance-20261004-r02-counterexample-0093",
  "pilot_number": "R0006-03543",
  "number": 3543,
  "family_id": "qf-counterexample-domain-restriction",
  "title": {
    "zh-Hant": "第 3543 題：限制論域後原反例的資格",
    "en": "Question 3543: Counterexample Eligibility after Domain Restriction"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261003-advance-20261004-r02-counterexample-0093",
    "batch_id": "batch-20261003-advance-20261004-r02-counterexample-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-counterexample"
    ],
    "title_zh": "限制論域後原反例的資格",
    "stem_zh": "明示有限論域 U={5, 6, 7}，整數條件 P(x)：x≥7，待檢驗全稱 ∀x∈U P(x)。c=5 原本是合法反例；改成明示子論域 V=∅ 後，c 還是合法反例嗎？",
    "answer": {
      "kind": "short_text",
      "value": "否",
      "explanation_zh": "c∈V=否；原論域述詞值=假。只有仍在 V 才保留見證資格；失去這個見證不自動判定新全稱為真。不外推到無限論域。"
    },
    "difficulty": {
      "band": "upper_secondary",
      "level": 3,
      "skills": [
        "限制論域後原反例的資格"
      ]
    },
    "variant": {
      "family_id": "qf-counterexample-domain-restriction",
      "seed": 202610042093,
      "parameters": {
        "U": [
          5,
          6,
          7
        ],
        "V": [],
        "predicate": {
          "kind": "ge",
          "value": 7
        },
        "c": 5,
        "operation": "restricted_witness",
        "semantic_template": "qf-counterexample-domain-restriction/{\"U\":[5,6,7],\"V\":[],\"c\":5,\"operation\":\"restricted_witness\",\"predicate\":{\"kind\":\"ge\",\"value\":7}}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_integer_rational_arithmetic",
        "producer_finite_enumeration",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication"
      ],
      "warning_zh": "僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "Codex independent peer (inherited model; exact runtime model not exposed)",
      "generated_at": "2026-10-03",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261003-advance-20261004-r02-counterexample-0093",
    "batch_id": "batch-20261003-advance-20261004-r02-counterexample-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "a269dc0829bc750ebe4b5bece4d78a7ec2fde7b63f9b920097da5ccc027114c3",
      "family_id": "qf-counterexample-domain-restriction",
      "seed": 202610042093,
      "semantic_template": "qf-counterexample-domain-restriction/{\"U\":[5,6,7],\"V\":[],\"c\":5,\"operation\":\"restricted_witness\",\"predicate\":{\"kind\":\"ge\",\"value\":7}}"
    },
    "fields": {
      "title": "Counterexample Eligibility after Domain Restriction",
      "stem": "Using classical truth semantics, the explicit finite integer domain is U={5, 6, 7} and P(x) means x≥7. The value c=5 is a valid counterexample to the original universal claim. After restricting to V=∅⊆U, is c still a valid counterexample?",
      "answer": {
        "kind": "short_text",
        "value": "No",
        "explanation": "The original predicate at c is false, and c does not belong to V, giving No. Loss of this witness does not itself establish truth of the new universal claim. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain."
      },
      "skills": [
        "Counterexample Eligibility after Domain Restriction"
      ],
      "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": "明示有限論域 U={5, 6, 7}，整數條件 P(x)：x≥7，待檢驗全稱 ∀x∈U P(x)。c=5 原本是合法反例；改成明示子論域 V=∅ 後，c 還是合法反例嗎？",
        "en": "Using classical truth semantics, the explicit finite integer domain is U={5, 6, 7} and P(x) means x≥7. The value c=5 is a valid counterexample to the original universal claim. After restricting to V=∅⊆U, is c still a valid counterexample?"
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "U={5, 6, 7};predicate=ge 7",
      "after": "a counterexample requires c∈U and P(c)=False",
      "relation": "implies",
      "rule_id": "finite_counterexample_domain",
      "conditions": [],
      "explanation": {
        "zh-Hant": "反例要同時符合論域資格與述詞失敗；域外整數即使使公式失敗，也不反駁此受限全稱。",
        "en": "A counterexample needs both domain membership and predicate failure;an outside integer does not refute this restricted universal statement even if its formula is false."
      },
      "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=5;predicate=ge 7",
      "after": "P(5)=否",
      "relation": "implies",
      "rule_id": "explicit_predicate_value",
      "conditions": [],
      "explanation": {
        "zh-Hant": "把這個整數完整代入宣告的條件；判定公式真值不會自行改變它的論域資格。",
        "en": "Substitute this integer into the declared condition;the formula's truth value does not change its domain eligibility."
      },
      "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": "c=5;c∈U=是;P(c)=否",
      "after": "original valid witness=是",
      "relation": "implies",
      "rule_id": "membership_and_predicate_failure",
      "conditions": [],
      "explanation": {
        "zh-Hant": "必須用合取，不能只檢查述詞假或只檢查在域內。",
        "en": "Use conjunction,not just predicate failure or domain membership alone."
      },
      "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": "V=∅⊆U;original witness=5",
      "after": "c∈V=否",
      "relation": "implies",
      "rule_id": "restricted_domain_witness",
      "conditions": [],
      "explanation": {
        "zh-Hant": "原反例只在仍屬新論域V時保留資格；失去這一反例不代表V內其他反例也消失。",
        "en": "The old counterexample remains eligible only if it is still in V;losing this witness does not remove every other possible counterexample in V."
      },
      "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": "the explicitly declared finite exercise",
      "after": "no inference about an unspecified infinite domain",
      "relation": "implies",
      "rule_id": "finite_logic_scope",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整枚舉可以決定這個有限全稱，但不把有限成功或無反例當作無限論域的普遍證明。",
        "en": "Complete enumeration decides this finite universal statement,but finite success or absence of counterexamples is not a universal proof over 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
    }
  ],
  "math_model": {
    "result": false,
    "original_witness_valid": true,
    "restricted_domain": []
  },
  "conclusion": {
    "canonical_value": "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."
  },
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    "original_candidate_line_sha256": "938df0c0b22a1147deb69e7d20e6a77b35654c47782f6361928f032d732a3714"
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      "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."
    },
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    "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.",
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}