{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261013-advance-20261018-r03-gcd-coprime-0013",
  "pilot_number": "R0062-31788",
  "number": 31788,
  "family_id": "qf-ext18-gcd-euclid-trace",
  "title": {
    "zh-Hant": "第 31788 題：最大公因數與互質",
    "en": "Question 31788: Exact Euclidean Remainder Trace and Natural GCD"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261013-advance-20261018-r03-gcd-coprime-0013",
    "batch_id": "batch-20261013-advance-20261018-r03-gcd-coprime-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-gcd-coprime"
    ],
    "title_zh": "最大公因數與互質",
    "stem_zh": "自然數gcd。操作=gcd-euclid-trace；資料={\"a\":119,\"b\":72}。請依 [a,b,euclid_rows,gcd] 的欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(119, 72, [[119, 72, 47], [72, 47, 25], [47, 25, 22], [25, 22, 3], [22, 3, 1], [3, 1, 0]], 1)",
      "explanation_zh": "擴展基礎核對=(119, 72, [[119, 72, 47], [72, 47, 25], [47, 25, 22], [25, 22, 3], [22, 3, 1], [3, 1, 0]], 1)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；gcd(0,0)=0；互質不表示兩數皆質數。"
    },
    "difficulty": {
      "band": "upper_secondary",
      "level": 2,
      "skills": [
        "最大公因數與互質"
      ]
    },
    "variant": {
      "family_id": "qf-ext18-gcd-euclid-trace",
      "seed": 202610183013,
      "parameters": {
        "a": 119,
        "b": 72,
        "domain": "Nat_gcd",
        "operation": "gcd-euclid-trace",
        "semantic_template": "qf-ext18-gcd-euclid-trace/{\"a\":119,\"b\":72}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_finite_arithmetic",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-13",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261013-advance-20261018-r03-gcd-coprime-0013",
    "batch_id": "batch-20261013-advance-20261018-r03-gcd-coprime-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "03c2067320059c5e40163021019a01b1e78923316ac0d3789ee95f9e3dbd642b",
      "family_id": "qf-ext18-gcd-euclid-trace",
      "seed": 202610183013,
      "semantic_template": "qf-ext18-gcd-euclid-trace/{\"a\":119,\"b\":72}"
    },
    "fields": {
      "title": "Exact Euclidean Remainder Trace and Natural GCD",
      "stem": "Use natural gcd with a=119, b=72. Give (a, b, ordered Euclid rows [x,y,x mod y] while y≠0, final gcd). Adopt gcd(0,0)=0.",
      "answer": {
        "kind": "expression",
        "value": "(119, 72, [[119, 72, 47], [72, 47, 25], [47, 25, 22], [25, 22, 3], [22, 3, 1], [3, 1, 0]], 1)",
        "explanation": "Never take a remainder with zero divisor. Terminate at y=0 and retain the final x; coprimality does not mean both inputs are prime. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (119, 72, [[119, 72, 47], [72, 47, 25], [47, 25, 22], [25, 22, 3], [22, 3, 1], [3, 1, 0]], 1)."
      },
      "skills": [
        "Exact Euclidean Remainder Trace and Natural GCD"
      ],
      "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": "自然數gcd。操作=gcd-euclid-trace；資料={\"a\":119,\"b\":72}。請依 [a,b,euclid_rows,gcd] 的欄序回答。",
        "en": "Use natural gcd with a=119, b=72. Give (a, b, ordered Euclid rows [x,y,x mod y] while y≠0, final gcd). Adopt gcd(0,0)=0."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source roles = [qf-ext18-gcd-euclid-trace, Nat_gcd, gcd-euclid-trace, no-form-field]",
      "after": "derived = [[a, b, ordered_Euclid_rows, gcd]]",
      "relation": "implies",
      "rule_id": "R62_EXACT_source_keys_domain_operation_form_AND_output_types",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原bank核欄位：BASE方程用form及論域，BASE事件／補集無operation，階乘無domain；ext18用local-hyphen，其他legacy用underscore。輸出型別不統一成tuple或fraction。",
        "en": "Check THIS bank interface:BASE equations use form/domain;BASE events/complement have no operation;factorials have no domain.ext18 uses local hyphens,other legacy operations underscores.Do not force every answer into a tuple or fraction."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向代回原輸入、欄位角色、論域及本步條件；一致性不表示結論能唯一恢復前提。有限核對、獨立審查、形式證明與發布分開標記。",
          "en": "Reverse-check the ORIGINAL inputs,field roles,domain and premises.Consistency does not imply unique recovery of premises.Keep finite checking,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "GCD Euclid row0 ORIGINAL quotient times y := mul(1, 72)",
      "after": "GCD Euclid row0 ORIGINAL quotient times y = 72",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "使用原操作數與明示次序作精確整數／有理計算，不用浮點近似。輸入界依題族：不把衍生值重驗成raw輸入，也不給BASE題庫另加SAFE或n64界。",
        "en": "Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 GCD Euclid row0 ORIGINAL quotient times y，不以小數近似替代。",
          "en": "Recompute GCD Euclid row0 ORIGINAL quotient times y with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "72"
        ],
        "output": "72",
        "label": "GCD Euclid row0 ORIGINAL quotient times y"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "GCD Euclid row0 ORIGINAL x minus qy := sub(119, 72)",
      "after": "GCD Euclid row0 ORIGINAL x minus qy = 47",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "使用原操作數與明示次序作精確整數／有理計算，不用浮點近似。輸入界依題族：不把衍生值重驗成raw輸入，也不給BASE題庫另加SAFE或n64界。",
        "en": "Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 GCD Euclid row0 ORIGINAL x minus qy，不以小數近似替代。",
          "en": "Recompute GCD Euclid row0 ORIGINAL x minus qy with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "119",
          "72"
        ],
        "output": "47",
        "label": "GCD Euclid row0 ORIGINAL x minus qy"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "source roles = [119, 72, 1]",
      "after": "derived = [72, 47, [119, 72, 47]]",
      "relation": "implies",
      "rule_id": "GCD_EUCLID_nonzero_divisor_exact_quotient_remainder_INVARIANT",
      "conditions": [],
      "explanation": {
        "zh-Hant": "y非0才求商餘，x=qy+r且0≤r<y。d同除x,y則同除y,r，反向由x=qy+r也成立，所以共同因數集合不變；正餘數嚴格下降，循環會終止。",
        "en": "With y≠0,divide to get x=qy+r and0≤r<y.A divisor common to x,y divides y,r,and the converse follows from x=qy+r,so common divisors are unchanged.Positive remainders strictly decrease,ensuring termination."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向代回原輸入、欄位角色、論域及本步條件；一致性不表示結論能唯一恢復前提。有限核對、獨立審查、形式證明與發布分開標記。",
          "en": "Reverse-check the ORIGINAL inputs,field roles,domain and premises.Consistency does not imply unique recovery of premises.Keep finite checking,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "GCD Euclid row1 ORIGINAL quotient times y := mul(1, 47)",
      "after": "GCD Euclid row1 ORIGINAL quotient times y = 47",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "使用原操作數與明示次序作精確整數／有理計算，不用浮點近似。輸入界依題族：不把衍生值重驗成raw輸入，也不給BASE題庫另加SAFE或n64界。",
        "en": "Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 GCD Euclid row1 ORIGINAL quotient times y，不以小數近似替代。",
          "en": "Recompute GCD Euclid row1 ORIGINAL quotient times y with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "47"
        ],
        "output": "47",
        "label": "GCD Euclid row1 ORIGINAL quotient times y"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "GCD Euclid row1 ORIGINAL x minus qy := sub(72, 47)",
      "after": "GCD Euclid row1 ORIGINAL x minus qy = 25",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "使用原操作數與明示次序作精確整數／有理計算，不用浮點近似。輸入界依題族：不把衍生值重驗成raw輸入，也不給BASE題庫另加SAFE或n64界。",
        "en": "Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 GCD Euclid row1 ORIGINAL x minus qy，不以小數近似替代。",
          "en": "Recompute GCD Euclid row1 ORIGINAL x minus qy with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "72",
          "47"
        ],
        "output": "25",
        "label": "GCD Euclid row1 ORIGINAL x minus qy"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "source roles = [72, 47, 1]",
      "after": "derived = [47, 25, [72, 47, 25]]",
      "relation": "implies",
      "rule_id": "GCD_EUCLID_nonzero_divisor_exact_quotient_remainder_INVARIANT",
      "conditions": [],
      "explanation": {
        "zh-Hant": "y非0才求商餘，x=qy+r且0≤r<y。d同除x,y則同除y,r，反向由x=qy+r也成立，所以共同因數集合不變；正餘數嚴格下降，循環會終止。",
        "en": "With y≠0,divide to get x=qy+r and0≤r<y.A divisor common to x,y divides y,r,and the converse follows from x=qy+r,so common divisors are unchanged.Positive remainders strictly decrease,ensuring termination."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向代回原輸入、欄位角色、論域及本步條件；一致性不表示結論能唯一恢復前提。有限核對、獨立審查、形式證明與發布分開標記。",
          "en": "Reverse-check the ORIGINAL inputs,field roles,domain and premises.Consistency does not imply unique recovery of premises.Keep finite checking,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "GCD Euclid row2 ORIGINAL quotient times y := mul(1, 25)",
      "after": "GCD Euclid row2 ORIGINAL quotient times y = 25",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "使用原操作數與明示次序作精確整數／有理計算，不用浮點近似。輸入界依題族：不把衍生值重驗成raw輸入，也不給BASE題庫另加SAFE或n64界。",
        "en": "Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 GCD Euclid row2 ORIGINAL quotient times y，不以小數近似替代。",
          "en": "Recompute GCD Euclid row2 ORIGINAL quotient times y with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "25"
        ],
        "output": "25",
        "label": "GCD Euclid row2 ORIGINAL quotient times y"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "GCD Euclid row2 ORIGINAL x minus qy := sub(47, 25)",
      "after": "GCD Euclid row2 ORIGINAL x minus qy = 22",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "使用原操作數與明示次序作精確整數／有理計算，不用浮點近似。輸入界依題族：不把衍生值重驗成raw輸入，也不給BASE題庫另加SAFE或n64界。",
        "en": "Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 GCD Euclid row2 ORIGINAL x minus qy，不以小數近似替代。",
          "en": "Recompute GCD Euclid row2 ORIGINAL x minus qy with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "47",
          "25"
        ],
        "output": "22",
        "label": "GCD Euclid row2 ORIGINAL x minus qy"
      }
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "source roles = [47, 25, 1]",
      "after": "derived = [25, 22, [47, 25, 22]]",
      "relation": "implies",
      "rule_id": "GCD_EUCLID_nonzero_divisor_exact_quotient_remainder_INVARIANT",
      "conditions": [],
      "explanation": {
        "zh-Hant": "y非0才求商餘，x=qy+r且0≤r<y。d同除x,y則同除y,r，反向由x=qy+r也成立，所以共同因數集合不變；正餘數嚴格下降，循環會終止。",
        "en": "With y≠0,divide to get x=qy+r and0≤r<y.A divisor common to x,y divides y,r,and the converse follows from x=qy+r,so common divisors are unchanged.Positive remainders strictly decrease,ensuring termination."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向代回原輸入、欄位角色、論域及本步條件；一致性不表示結論能唯一恢復前提。有限核對、獨立審查、形式證明與發布分開標記。",
          "en": "Reverse-check the ORIGINAL inputs,field roles,domain and premises.Consistency does not imply unique recovery of premises.Keep finite checking,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "GCD Euclid row3 ORIGINAL quotient times y := mul(1, 22)",
      "after": "GCD Euclid row3 ORIGINAL quotient times y = 22",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "使用原操作數與明示次序作精確整數／有理計算，不用浮點近似。輸入界依題族：不把衍生值重驗成raw輸入，也不給BASE題庫另加SAFE或n64界。",
        "en": "Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 GCD Euclid row3 ORIGINAL quotient times y，不以小數近似替代。",
          "en": "Recompute GCD Euclid row3 ORIGINAL quotient times y with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "22"
        ],
        "output": "22",
        "label": "GCD Euclid row3 ORIGINAL quotient times y"
      }
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "GCD Euclid row3 ORIGINAL x minus qy := sub(25, 22)",
      "after": "GCD Euclid row3 ORIGINAL x minus qy = 3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "使用原操作數與明示次序作精確整數／有理計算，不用浮點近似。輸入界依題族：不把衍生值重驗成raw輸入，也不給BASE題庫另加SAFE或n64界。",
        "en": "Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 GCD Euclid row3 ORIGINAL x minus qy，不以小數近似替代。",
          "en": "Recompute GCD Euclid row3 ORIGINAL x minus qy with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "25",
          "22"
        ],
        "output": "3",
        "label": "GCD Euclid row3 ORIGINAL x minus qy"
      }
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "source roles = [25, 22, 1]",
      "after": "derived = [22, 3, [25, 22, 3]]",
      "relation": "implies",
      "rule_id": "GCD_EUCLID_nonzero_divisor_exact_quotient_remainder_INVARIANT",
      "conditions": [],
      "explanation": {
        "zh-Hant": "y非0才求商餘，x=qy+r且0≤r<y。d同除x,y則同除y,r，反向由x=qy+r也成立，所以共同因數集合不變；正餘數嚴格下降，循環會終止。",
        "en": "With y≠0,divide to get x=qy+r and0≤r<y.A divisor common to x,y divides y,r,and the converse follows from x=qy+r,so common divisors are unchanged.Positive remainders strictly decrease,ensuring termination."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向代回原輸入、欄位角色、論域及本步條件；一致性不表示結論能唯一恢復前提。有限核對、獨立審查、形式證明與發布分開標記。",
          "en": "Reverse-check the ORIGINAL inputs,field roles,domain and premises.Consistency does not imply unique recovery of premises.Keep finite checking,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "GCD Euclid row4 ORIGINAL quotient times y := mul(7, 3)",
      "after": "GCD Euclid row4 ORIGINAL quotient times y = 21",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "使用原操作數與明示次序作精確整數／有理計算，不用浮點近似。輸入界依題族：不把衍生值重驗成raw輸入，也不給BASE題庫另加SAFE或n64界。",
        "en": "Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 GCD Euclid row4 ORIGINAL quotient times y，不以小數近似替代。",
          "en": "Recompute GCD Euclid row4 ORIGINAL quotient times y with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "7",
          "3"
        ],
        "output": "21",
        "label": "GCD Euclid row4 ORIGINAL quotient times y"
      }
    },
    {
      "id": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "GCD Euclid row4 ORIGINAL x minus qy := sub(22, 21)",
      "after": "GCD Euclid row4 ORIGINAL x minus qy = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "使用原操作數與明示次序作精確整數／有理計算，不用浮點近似。輸入界依題族：不把衍生值重驗成raw輸入，也不給BASE題庫另加SAFE或n64界。",
        "en": "Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 GCD Euclid row4 ORIGINAL x minus qy，不以小數近似替代。",
          "en": "Recompute GCD Euclid row4 ORIGINAL x minus qy with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "22",
          "21"
        ],
        "output": "1",
        "label": "GCD Euclid row4 ORIGINAL x minus qy"
      }
    },
    {
      "id": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "source roles = [22, 3, 7]",
      "after": "derived = [21, 1, [22, 3, 1]]",
      "relation": "implies",
      "rule_id": "GCD_EUCLID_nonzero_divisor_exact_quotient_remainder_INVARIANT",
      "conditions": [],
      "explanation": {
        "zh-Hant": "y非0才求商餘，x=qy+r且0≤r<y。d同除x,y則同除y,r，反向由x=qy+r也成立，所以共同因數集合不變；正餘數嚴格下降，循環會終止。",
        "en": "With y≠0,divide to get x=qy+r and0≤r<y.A divisor common to x,y divides y,r,and the converse follows from x=qy+r,so common divisors are unchanged.Positive remainders strictly decrease,ensuring termination."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向代回原輸入、欄位角色、論域及本步條件；一致性不表示結論能唯一恢復前提。有限核對、獨立審查、形式證明與發布分開標記。",
          "en": "Reverse-check the ORIGINAL inputs,field roles,domain and premises.Consistency does not imply unique recovery of premises.Keep finite checking,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S017",
      "uses": [
        "G001",
        "S016"
      ],
      "before": "GCD Euclid row5 ORIGINAL quotient times y := mul(3, 1)",
      "after": "GCD Euclid row5 ORIGINAL quotient times y = 3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "使用原操作數與明示次序作精確整數／有理計算，不用浮點近似。輸入界依題族：不把衍生值重驗成raw輸入，也不給BASE題庫另加SAFE或n64界。",
        "en": "Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 GCD Euclid row5 ORIGINAL quotient times y，不以小數近似替代。",
          "en": "Recompute GCD Euclid row5 ORIGINAL quotient times y with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "3",
          "1"
        ],
        "output": "3",
        "label": "GCD Euclid row5 ORIGINAL quotient times y"
      }
    },
    {
      "id": "S018",
      "uses": [
        "G001",
        "S017"
      ],
      "before": "GCD Euclid row5 ORIGINAL x minus qy := sub(3, 3)",
      "after": "GCD Euclid row5 ORIGINAL x minus qy = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "使用原操作數與明示次序作精確整數／有理計算，不用浮點近似。輸入界依題族：不把衍生值重驗成raw輸入，也不給BASE題庫另加SAFE或n64界。",
        "en": "Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 GCD Euclid row5 ORIGINAL x minus qy，不以小數近似替代。",
          "en": "Recompute GCD Euclid row5 ORIGINAL x minus qy with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "3",
          "3"
        ],
        "output": "0",
        "label": "GCD Euclid row5 ORIGINAL x minus qy"
      }
    },
    {
      "id": "S019",
      "uses": [
        "G001",
        "S018"
      ],
      "before": "source roles = [3, 1, 3]",
      "after": "derived = [3, 0, [3, 1, 0]]",
      "relation": "implies",
      "rule_id": "GCD_EUCLID_nonzero_divisor_exact_quotient_remainder_INVARIANT",
      "conditions": [],
      "explanation": {
        "zh-Hant": "y非0才求商餘，x=qy+r且0≤r<y。d同除x,y則同除y,r，反向由x=qy+r也成立，所以共同因數集合不變；正餘數嚴格下降，循環會終止。",
        "en": "With y≠0,divide to get x=qy+r and0≤r<y.A divisor common to x,y divides y,r,and the converse follows from x=qy+r,so common divisors are unchanged.Positive remainders strictly decrease,ensuring termination."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向代回原輸入、欄位角色、論域及本步條件；一致性不表示結論能唯一恢復前提。有限核對、獨立審查、形式證明與發布分開標記。",
          "en": "Reverse-check the ORIGINAL inputs,field roles,domain and premises.Consistency does not imply unique recovery of premises.Keep finite checking,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S020",
      "uses": [
        "G001",
        "S019"
      ],
      "before": "source roles = [119, 72, [[119, 72, 47], [72, 47, 25], [47, 25, 22], [25, 22, 3], [22, 3, 1], [3, 1, 0]]]",
      "after": "derived = [1, 0]",
      "relation": "implies",
      "rule_id": "GCD_TERMINAL_last_x_ZEROsecond_and_explicit_gcd00_convention",
      "conditions": [],
      "explanation": {
        "zh-Hant": "到第二值0時保留最後x：非零時gcd(x,0)=x，連同上面雙向共同因數不變式得到原gcd。原(0,0)採明示gcd=0慣例，不宣稱0是最大的正因數。",
        "en": "At second value0 retain the last x.For nonzero x,gcd(x,0)=x;the two-way common-divisor invariant transfers this to the original pair.At(0,0)use the EXPLICIT convention gcd=0,not a claim that0 is the largest positive divisor."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向代回原輸入、欄位角色、論域及本步條件；一致性不表示結論能唯一恢復前提。有限核對、獨立審查、形式證明與發布分開標記。",
          "en": "Reverse-check the ORIGINAL inputs,field roles,domain and premises.Consistency does not imply unique recovery of premises.Keep finite checking,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "result": [
      119,
      72,
      [
        [
          119,
          72,
          47
        ],
        [
          72,
          47,
          25
        ],
        [
          47,
          25,
          22
        ],
        [
          25,
          22,
          3
        ],
        [
          22,
          3,
          1
        ],
        [
          3,
          1,
          0
        ]
      ],
      1
    ]
  },
  "conclusion": {
    "canonical_value": "(119, 72, [[119, 72, 47], [72, 47, 25], [47, 25, 22], [25, 22, 3], [22, 3, 1], [3, 1, 0]], 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-20261013-advance-20261018-r03-gcd-coprime-foundations",
      "batch_path": "public/data/questions/batches/batch-20261013-advance-20261018-r03-gcd-coprime-foundations/questions.jsonl",
      "batch_sha256": "42ddb0f9d2780c613fbb13b7c105532a263effd3cc83d5abb31fa93c39b1f6e1",
      "original_line": 13,
      "original_line_sha256": "981efe0b8cae303ca4e68e79c01a26ebf78741994a759189e6933c2e3e358719",
      "question_compact_sha256": "03c2067320059c5e40163021019a01b1e78923316ac0d3789ee95f9e3dbd642b",
      "english_row_compact_sha256": "b8513d6f46106343bd4edd24daf5b87cbe4d0f7bed2fcbf007a9cdc7a3069e2b",
      "english_manifest_path": "public/data/i18n/editions/batch-20261013-advance-20261018-r03-gcd-coprime-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261013-advance-20261018-r03-gcd-coprime-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-gcd-coprime": {
        "sha256": "a8f5934d6354ea07a83057c61f045034e2d964850bed4759aedd8721848b3b9b",
        "english_sha256": "9a16e24c872347b1ca2aff7ffd64e3465452e78b46eabfda5a844f785741500e",
        "original_path": "public/data/mko/gcd-coprime.json",
        "english_path": "public/data/i18n/editions/mko-gcd-coprime.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "59d384ca3d958b27165f4295ff96efb155e7b12d3fb70bcdd0ed68148513c05a",
    "original_candidate_sha256": "be2745d7487dbaa7da0afd9e2d9880c5da22de1027aa656b38b2760a5f0a0aef",
    "original_candidate_line_sha256": "be2745d7487dbaa7da0afd9e2d9880c5da22de1027aa656b38b2760a5f0a0aef"
  },
  "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-20261026-011-p63",
    "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": "963099ffc9f754069fdce6792949a85ed350472465eaceb3038fa8011cf672bd",
      "question_first_witness_sha256": "d7655324c5d11528e1af90b757384cebd3313e19dc5f6f5b9b8ea7624fb06f6f"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 20,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "2f95594d5d680663e08914a51a001e3c3bbe77a1b6d5d91499b5159f08ce3b9e",
      "arithmetic_steps": 12
    },
    "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."
  }
}