{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261003-advance-20261004-r02-counterexample-0071",
  "pilot_number": "R0006-03521",
  "number": 3521,
  "family_id": "qf-counterexample-candidate-set",
  "title": {
    "zh-Hant": "第 3521 題：篩出候選集合中的合法反例",
    "en": "Question 3521: Valid Counterexamples in a Candidate Set"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261003-advance-20261004-r02-counterexample-0071",
    "batch_id": "batch-20261003-advance-20261004-r02-counterexample-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-counterexample"
    ],
    "title_zh": "篩出候選集合中的合法反例",
    "stem_zh": "明示有限論域 U={8, 9, 11}，整數條件 P(x)：x≥9，待檢驗全稱 ∀x∈U P(x)。候選集合 C={6, 8, 9, 11, 15} 中，哪些元素是合法反例？輸出遞增且元素只列一次的集合。",
    "answer": {
      "kind": "expression",
      "value": "{8}",
      "explanation_zh": "合法候選反例={8}。需同時在 U、在 C，且使述詞失敗；域外值不具資格。不外推到無限論域。"
    },
    "difficulty": {
      "band": "upper_secondary",
      "level": 3,
      "skills": [
        "篩出候選集合中的合法反例"
      ]
    },
    "variant": {
      "family_id": "qf-counterexample-candidate-set",
      "seed": 202610042071,
      "parameters": {
        "U": [
          8,
          9,
          11
        ],
        "predicate": {
          "kind": "ge",
          "value": 9
        },
        "C": [
          6,
          8,
          9,
          11,
          15
        ],
        "operation": "candidate_set",
        "semantic_template": "qf-counterexample-candidate-set/{\"C\":[6,8,9,11,15],\"U\":[8,9,11],\"operation\":\"candidate_set\",\"predicate\":{\"kind\":\"ge\",\"value\":9}}"
      }
    },
    "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-0071",
    "batch_id": "batch-20261003-advance-20261004-r02-counterexample-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "4fa3fefc6e564a0d4d3b4dfcfea17ce4e58a2417aaaea120a85325dd00754973",
      "family_id": "qf-counterexample-candidate-set",
      "seed": 202610042071,
      "semantic_template": "qf-counterexample-candidate-set/{\"C\":[6,8,9,11,15],\"U\":[8,9,11],\"operation\":\"candidate_set\",\"predicate\":{\"kind\":\"ge\",\"value\":9}}"
    },
    "fields": {
      "title": "Valid Counterexamples in a Candidate Set",
      "stem": "Using classical truth semantics, the explicit finite integer domain is U={8, 9, 11} and P(x) means x≥9. Which members of C={6, 8, 9, 11, 15} are valid counterexamples to ∀x∈U P(x)? Return the set in increasing order with no duplicates.",
      "answer": {
        "kind": "expression",
        "value": "{8}",
        "explanation": "Require membership in both U and C and failure of P. The valid candidate counterexample set is {8}; outside values are ineligible. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain."
      },
      "skills": [
        "Valid Counterexamples in a 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": "明示有限論域 U={8, 9, 11}，整數條件 P(x)：x≥9，待檢驗全稱 ∀x∈U P(x)。候選集合 C={6, 8, 9, 11, 15} 中，哪些元素是合法反例？輸出遞增且元素只列一次的集合。",
        "en": "Using classical truth semantics, the explicit finite integer domain is U={8, 9, 11} and P(x) means x≥9. Which members of C={6, 8, 9, 11, 15} are valid counterexamples to ∀x∈U P(x)? Return the set in increasing order with no duplicates."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "U={8, 9, 11};predicate=ge 9",
      "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=8;predicate=ge 9",
      "after": "P(8)=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": "x=9;predicate=ge 9",
      "after": "P(9)=是",
      "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": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "x=11;predicate=ge 9",
      "after": "P(11)=是",
      "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": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "[[8, 否], [9, 是], [11, 是]]",
      "after": "all finite-domain failures={8}",
      "relation": "implies",
      "rule_id": "complete_finite_failure_inventory",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐一涵蓋整個明示有限論域；空U的全稱為空真且沒有反例，不外推到更大或無限論域。",
        "en": "Cover the entire declared finite domain;the universal statement on empty U is vacuously true and has no counterexample.This does not extend to larger or infinite domains."
      },
      "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": "candidate=6;c∈U=否",
      "after": "legal counterexample=否",
      "relation": "implies",
      "rule_id": "candidate_counterexample_filter",
      "conditions": [],
      "explanation": {
        "zh-Hant": "候選要同時在C與U內並使述詞失敗；域外述詞留未取值，不假裝它是全稱的反例。",
        "en": "A selected candidate must lie in both C and U and fail the predicate;an outside-domain predicate is left unevaluated for this inventory."
      },
      "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": "candidate=8;c∈U=是",
      "after": "legal counterexample=是",
      "relation": "implies",
      "rule_id": "candidate_counterexample_filter",
      "conditions": [],
      "explanation": {
        "zh-Hant": "候選要同時在C與U內並使述詞失敗；域外述詞留未取值，不假裝它是全稱的反例。",
        "en": "A selected candidate must lie in both C and U and fail the predicate;an outside-domain predicate is left unevaluated for this inventory."
      },
      "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": "candidate=9;c∈U=是",
      "after": "legal counterexample=否",
      "relation": "implies",
      "rule_id": "candidate_counterexample_filter",
      "conditions": [],
      "explanation": {
        "zh-Hant": "候選要同時在C與U內並使述詞失敗；域外述詞留未取值，不假裝它是全稱的反例。",
        "en": "A selected candidate must lie in both C and U and fail the predicate;an outside-domain predicate is left unevaluated for this inventory."
      },
      "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=11;c∈U=是",
      "after": "legal counterexample=否",
      "relation": "implies",
      "rule_id": "candidate_counterexample_filter",
      "conditions": [],
      "explanation": {
        "zh-Hant": "候選要同時在C與U內並使述詞失敗；域外述詞留未取值，不假裝它是全稱的反例。",
        "en": "A selected candidate must lie in both C and U and fail the predicate;an outside-domain predicate is left unevaluated for this inventory."
      },
      "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=15;c∈U=否",
      "after": "legal counterexample=否",
      "relation": "implies",
      "rule_id": "candidate_counterexample_filter",
      "conditions": [],
      "explanation": {
        "zh-Hant": "候選要同時在C與U內並使述詞失敗；域外述詞留未取值，不假裝它是全稱的反例。",
        "en": "A selected candidate must lie in both C and U and fail the predicate;an outside-domain predicate is left unevaluated for this inventory."
      },
      "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": "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": [
      8
    ],
    "failures": [
      8
    ],
    "candidate_rows": [
      [
        6,
        false,
        null,
        false
      ],
      [
        8,
        true,
        false,
        true
      ],
      [
        9,
        true,
        true,
        false
      ],
      [
        11,
        true,
        true,
        false
      ],
      [
        15,
        false,
        null,
        false
      ]
    ]
  },
  "conclusion": {
    "canonical_value": "{8}",
    "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",
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      "batch_sha256": "1a6e9884f6c46dbd9ed0cfcc878b6d61d6fc0d37d3f15f0e18be439db59944a9",
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      "original_line_sha256": "9dd115cdf548459562f3ce2e55b2d7081fc4a027ace4cf7d9dab8b2e20fbc9a7",
      "question_compact_sha256": "4fa3fefc6e564a0d4d3b4dfcfea17ce4e58a2417aaaea120a85325dd00754973",
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      "english_manifest_path": "public/data/i18n/editions/batch-20261003-advance-20261004-r02-counterexample-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261003-advance-20261004-r02-counterexample-foundations.en.jsonl"
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    "mko_bindings": {
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        "english_sha256": "8fb486a8d29e7dd741f3fc050973e1897157bfa5f03ee86c37e9b8e1b728a8e4",
        "original_path": "public/data/mko/counterexample.json",
        "english_path": "public/data/i18n/editions/mko-counterexample.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "6bac5f83f28cdd9c992a469306bfdf5c5415de17d11847cb9b17b4d13934c7e6",
    "original_candidate_sha256": "3202a1223c627859f7bec216390f1f74a3b53dca36f41eb89aba49d2ee366362",
    "original_candidate_line_sha256": "3202a1223c627859f7bec216390f1f74a3b53dca36f41eb89aba49d2ee366362"
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  "historical_source_state": {
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      "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-p7",
    "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": "568eefd71709ac3dc16a0d5b3b4ad3e83376a8ef300c9e5493eee0721500b29d",
      "question_first_witness_sha256": "d964b91bedb9d4918bbc055f567762a09f5a58af6c876268e719d5100332d855"
    },
    "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": "5c431c2a48febe90ffa5eed8f79a3170149e22a7792666a3583e8c534d76ebe7",
      "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."
  }
}