{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261014-advance-20261105-r03-euclidean-algorithm-0017",
  "pilot_number": "R0036-18842",
  "number": 18842,
  "family_id": "qf-mod36-euclid-division-chain",
  "title": {
    "zh-Hant": "第 18842 題：Euclidean演算法",
    "en": "Question 18842: Exact Quotient-Remainder Steps and Strict Descent Are Both Required"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261014-advance-20261105-r03-euclidean-algorithm-0017",
    "batch_id": "batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-euclidean-algorithm"
    ],
    "title_zh": "Euclidean演算法",
    "stem_zh": "exact integer quotient-remainder chain與嚴格餘數下降。操作=euclid-division-chain；資料={\"a\":1592,\"b\":113}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "([[1592, 113, 14, 10], [113, 10, 11, 3], [10, 3, 3, 1], [3, 1, 3, 0]], 1, 是, exact_quotient_remainder_and_strict_descent)",
      "explanation_zh": "Wave5第五層核對=([[1592, 113, 14, 10], [113, 10, 11, 3], [10, 3, 3, 1], [3, 1, 3, 0]], 1, 是, exact_quotient_remainder_and_strict_descent)；依完整明示資料獨立重算；gcd invariant、終止與非負gcd分別核對且不執行除以零；Bezout係數由extended Euclidean回代並驗ax+by=gcd；gcd(0,0)=0 convention明示。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 2,
      "skills": [
        "Euclidean演算法"
      ]
    },
    "variant": {
      "family_id": "qf-mod36-euclid-division-chain",
      "seed": 202611053017,
      "parameters": {
        "a": 1592,
        "b": 113,
        "domain": "Exact_integer_Euclidean_chain",
        "operation": "euclid-division-chain",
        "semantic_template": "qf-mod36-euclid-division-chain/{\"a\":1592,\"b\":113}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_finite_or_symbolic_model",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "候選須主端重算；不構成人工審定、有限取樣全稱證明、V與dual典範識別、Hamilton結論、一般continuous-bijection捷徑、統計性質、strong law、CLT、Lean或Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-14",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261014-advance-20261105-r03-euclidean-algorithm-0017",
    "batch_id": "batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "58ef397a5fe4396edb56675fc869c95ad6d5426e9ce8eb0696de7abdee6bd929",
      "family_id": "qf-mod36-euclid-division-chain",
      "seed": 202611053017,
      "semantic_template": "qf-mod36-euclid-division-chain/{\"a\":1592,\"b\":113}"
    },
    "fields": {
      "title": "Exact Quotient-Remainder Steps and Strict Descent Are Both Required",
      "stem": "Use original integers a=1592, b=113. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (division_chain, gcd, chain_valid, certificate).",
      "answer": {
        "kind": "expression",
        "value": "([[1592, 113, 14, 10], [113, 10, 11, 3], [10, 3, 3, 1], [3, 1, 3, 0]], 1, Yes, exact_quotient_remainder_and_strict_descent)",
        "explanation": "Return every division row including the final zero remainder, then the nonnegative gcd and chain certificate. The ordinary family requires b≠0; zero-divisor termination is not an extra division step. The exact result, retaining the requested order, is ([[1592, 113, 14, 10], [113, 10, 11, 3], [10, 3, 3, 1], [3, 1, 3, 0]], 1, Yes, exact_quotient_remainder_and_strict_descent)."
      },
      "skills": [
        "Exact Quotient-Remainder Steps and Strict Descent Are Both Required"
      ],
      "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": "exact integer quotient-remainder chain與嚴格餘數下降。操作=euclid-division-chain；資料={\"a\":1592,\"b\":113}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
        "en": "Use original integers a=1592, b=113. Execute the Euclidean chain on their absolute values; each row [left, right, quotient, remainder] satisfies left=right*quotient+remainder with 0≤remainder<right. Stop before a zero divisor. Restore original signs for Bezout coefficients; use the explicit gcd(0,0)=0 convention only in its boundary family. Give (division_chain, gcd, chain_valid, certificate)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source operation=euclid-division-chain;original snapshot and mathematics retained",
      "after": "explicit ordered source output fields=[division_chain, gcd, chain_valid, certificate]",
      "relation": "implies",
      "rule_id": "source_ordered_interface_explicit_without_javascript",
      "conditions": [],
      "explanation": {
        "zh-Hant": "原題歷史介面標記仍在快照；這裡直接給原始欄位順序，主文不需JavaScript或外部表。描述欄位不自動證成任何數學聲稱。",
        "en": "Keep the historical interface marker in the original snapshot while listing the original ordered fields here.No JavaScript or external table is needed for the main text,and field labels are not mathematical evidence."
      },
      "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": "original signed inputs a=1592,b=113",
      "after": "initial nonnegative chain values=[1592, 113];gcd(a,b)=gcd(abs(a),abs(b));never divide by a zero right input",
      "relation": "implies",
      "rule_id": "euclid_signed_inputs_absolute_chain_scope",
      "conditions": [],
      "explanation": {
        "zh-Hant": "公因數的整除不受符號改變影響；gcd取非負。一般chain/invariant要求b非零，零邊界另保留(0,0)約定，不擅自套該限制。",
        "en": "Divisibility is unaffected by input signs,and the gcd is nonnegative.Ordinary chain/invariant families require b≠0;the zero boundary separately retains its(0,0)convention."
      },
      "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": "Euclid step0 exact ratio := div(1592, 113)",
      "after": "Euclid step0 exact ratio = 1592/113",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。",
        "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euclid step0 exact ratio，不以小數近似替代。",
          "en": "Recompute Euclid step0 exact ratio with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "1592",
          "113"
        ],
        "output": "1592/113",
        "label": "Euclid step0 exact ratio"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "Euclid step0 quotient times divisor := mul(14, 113)",
      "after": "Euclid step0 quotient times divisor = 1582",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。",
        "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euclid step0 quotient times divisor，不以小數近似替代。",
          "en": "Recompute Euclid step0 quotient times divisor with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "14",
          "113"
        ],
        "output": "1582",
        "label": "Euclid step0 quotient times divisor"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "Euclid step0 remainder := sub(1592, 1582)",
      "after": "Euclid step0 remainder = 10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。",
        "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euclid step0 remainder，不以小數近似替代。",
          "en": "Recompute Euclid step0 remainder with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "1592",
          "1582"
        ],
        "output": "10",
        "label": "Euclid step0 remainder"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "step=0;left=1592;positive right=113;exact ratio=1592/113;floor quotient=14",
      "after": "left=right*q+r with r=10 in[0,113);gcd pair=[1, 1]",
      "relation": "implies",
      "rule_id": "euclid_each_division_common_divisors_and_strict_descent",
      "conditions": [],
      "explanation": {
        "zh-Hant": "任意d同時整除left,right，等價整除right,left−q·right，所以gcd不變。下一個非負right是嚴格較小r，故不能無限下降；這是與不變性分開的終止理由。",
        "en": "An integer d divides both left and right iff it divides right and left−q*right,so the gcd is unchanged.The next nonnegative right is the strictly smaller remainder,so infinite descent is impossible;termination is separate from invariance."
      },
      "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": "Euclid step1 exact ratio := div(113, 10)",
      "after": "Euclid step1 exact ratio = 113/10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。",
        "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euclid step1 exact ratio，不以小數近似替代。",
          "en": "Recompute Euclid step1 exact ratio with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "113",
          "10"
        ],
        "output": "113/10",
        "label": "Euclid step1 exact ratio"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "Euclid step1 quotient times divisor := mul(11, 10)",
      "after": "Euclid step1 quotient times divisor = 110",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。",
        "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euclid step1 quotient times divisor，不以小數近似替代。",
          "en": "Recompute Euclid step1 quotient times divisor with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "11",
          "10"
        ],
        "output": "110",
        "label": "Euclid step1 quotient times divisor"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "Euclid step1 remainder := sub(113, 110)",
      "after": "Euclid step1 remainder = 3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。",
        "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euclid step1 remainder，不以小數近似替代。",
          "en": "Recompute Euclid step1 remainder with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "113",
          "110"
        ],
        "output": "3",
        "label": "Euclid step1 remainder"
      }
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "step=1;left=113;positive right=10;exact ratio=113/10;floor quotient=11",
      "after": "left=right*q+r with r=3 in[0,10);gcd pair=[1, 1]",
      "relation": "implies",
      "rule_id": "euclid_each_division_common_divisors_and_strict_descent",
      "conditions": [],
      "explanation": {
        "zh-Hant": "任意d同時整除left,right，等價整除right,left−q·right，所以gcd不變。下一個非負right是嚴格較小r，故不能無限下降；這是與不變性分開的終止理由。",
        "en": "An integer d divides both left and right iff it divides right and left−q*right,so the gcd is unchanged.The next nonnegative right is the strictly smaller remainder,so infinite descent is impossible;termination is separate from invariance."
      },
      "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": "Euclid step2 exact ratio := div(10, 3)",
      "after": "Euclid step2 exact ratio = 10/3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。",
        "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euclid step2 exact ratio，不以小數近似替代。",
          "en": "Recompute Euclid step2 exact ratio with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "10",
          "3"
        ],
        "output": "10/3",
        "label": "Euclid step2 exact ratio"
      }
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "Euclid step2 quotient times divisor := mul(3, 3)",
      "after": "Euclid step2 quotient times divisor = 9",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。",
        "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euclid step2 quotient times divisor，不以小數近似替代。",
          "en": "Recompute Euclid step2 quotient times divisor with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "3",
          "3"
        ],
        "output": "9",
        "label": "Euclid step2 quotient times divisor"
      }
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "Euclid step2 remainder := sub(10, 9)",
      "after": "Euclid step2 remainder = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。",
        "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euclid step2 remainder，不以小數近似替代。",
          "en": "Recompute Euclid step2 remainder with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "10",
          "9"
        ],
        "output": "1",
        "label": "Euclid step2 remainder"
      }
    },
    {
      "id": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "step=2;left=10;positive right=3;exact ratio=10/3;floor quotient=3",
      "after": "left=right*q+r with r=1 in[0,3);gcd pair=[1, 1]",
      "relation": "implies",
      "rule_id": "euclid_each_division_common_divisors_and_strict_descent",
      "conditions": [],
      "explanation": {
        "zh-Hant": "任意d同時整除left,right，等價整除right,left−q·right，所以gcd不變。下一個非負right是嚴格較小r，故不能無限下降；這是與不變性分開的終止理由。",
        "en": "An integer d divides both left and right iff it divides right and left−q*right,so the gcd is unchanged.The next nonnegative right is the strictly smaller remainder,so infinite descent is impossible;termination is separate from invariance."
      },
      "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": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "Euclid step3 exact ratio := div(3, 1)",
      "after": "Euclid step3 exact ratio = 3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。",
        "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euclid step3 exact ratio，不以小數近似替代。",
          "en": "Recompute Euclid step3 exact ratio with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "3",
          "1"
        ],
        "output": "3",
        "label": "Euclid step3 exact ratio"
      }
    },
    {
      "id": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "Euclid step3 quotient times divisor := mul(3, 1)",
      "after": "Euclid step3 quotient times divisor = 3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。",
        "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euclid step3 quotient times divisor，不以小數近似替代。",
          "en": "Recompute Euclid step3 quotient times divisor with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "3",
          "1"
        ],
        "output": "3",
        "label": "Euclid step3 quotient times divisor"
      }
    },
    {
      "id": "S017",
      "uses": [
        "G001",
        "S016"
      ],
      "before": "Euclid step3 remainder := sub(3, 3)",
      "after": "Euclid step3 remainder = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原資料角色及前面已核中間量精確代值，保留索引、正負號與所在空間。",
        "en": "Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euclid step3 remainder，不以小數近似替代。",
          "en": "Recompute Euclid step3 remainder with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "3",
          "3"
        ],
        "output": "0",
        "label": "Euclid step3 remainder"
      }
    },
    {
      "id": "S018",
      "uses": [
        "G001",
        "S017"
      ],
      "before": "step=3;left=3;positive right=1;exact ratio=3;floor quotient=3",
      "after": "left=right*q+r with r=0 in[0,1);gcd pair=[1, 1]",
      "relation": "implies",
      "rule_id": "euclid_each_division_common_divisors_and_strict_descent",
      "conditions": [],
      "explanation": {
        "zh-Hant": "任意d同時整除left,right，等價整除right,left−q·right，所以gcd不變。下一個非負right是嚴格較小r，故不能無限下降；這是與不變性分開的終止理由。",
        "en": "An integer d divides both left and right iff it divides right and left−q*right,so the gcd is unchanged.The next nonnegative right is the strictly smaller remainder,so infinite descent is impossible;termination is separate from invariance."
      },
      "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": "S019",
      "uses": [
        "G001",
        "S018"
      ],
      "before": "complete division rows=[[1592, 113, 14, 10], [113, 10, 11, 3], [10, 3, 3, 1], [3, 1, 3, 0]];terminal pair=[1, 0]",
      "after": "gcd(a,b)=gcd(1,0)=1;all divisions used positive divisors;nonnegative gcd convention",
      "relation": "implies",
      "rule_id": "euclid_terminal_gcd_not_extra_zero_division",
      "conditions": [],
      "explanation": {
        "zh-Hant": "終止時所有共同因數等於terminal left的因數，非負最大公因數就是left；(0,0)的0另由明示約定給出。不新增除以零的虛假一列。",
        "en": "At termination common divisors are those of the terminal left value,giving its nonnegative gcd.The(0,0)value0 is an explicit separate convention.Do not append a fictitious division by zero."
      },
      "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": [
      [
        [
          1592,
          113,
          14,
          10
        ],
        [
          113,
          10,
          11,
          3
        ],
        [
          10,
          3,
          3,
          1
        ],
        [
          3,
          1,
          3,
          0
        ]
      ],
      1,
      true,
      "exact_quotient_remainder_and_strict_descent"
    ]
  },
  "conclusion": {
    "canonical_value": "([[1592, 113, 14, 10], [113, 10, 11, 3], [10, 3, 3, 1], [3, 1, 3, 0]], 1, 是, exact_quotient_remainder_and_strict_descent)",
    "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-20261014-advance-20261105-r03-euclidean-algorithm-foundations",
      "batch_path": "public/data/questions/batches/batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations/questions.jsonl",
      "batch_sha256": "d39ff24de9e0227427922e795a2ba626435ae1ca24019ebf7583df6a77cb6d6b",
      "original_line": 17,
      "original_line_sha256": "171dbfcec8d943ca986bade28dd38643753a0ddb492117aa669725bd2ba87f81",
      "question_compact_sha256": "58ef397a5fe4396edb56675fc869c95ad6d5426e9ce8eb0696de7abdee6bd929",
      "english_row_compact_sha256": "e10e4d57254dc4cf41fbc85cf0d2eba0cb703230ecfd6e211a62d24be677c7db",
      "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261105-r03-euclidean-algorithm-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-euclidean-algorithm": {
        "sha256": "eadc65605f926f582e90694e2b3cd2fad1b79c8f39d0f5ecca8962c1ef2e95bf",
        "english_sha256": "b75e7f75026cacf6d854568514b08487eb437112b1c0bc09062dbe1379b77f3c",
        "original_path": "public/data/mko/euclidean-algorithm.json",
        "english_path": "public/data/i18n/editions/mko-euclidean-algorithm.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "49e5d7674cefa01298c2386543f777d2bdfc99446b4027156b80142f10031994",
    "original_candidate_sha256": "1a22c26d43540d9d102e0c10f71b03d39ecbb9544f1a213819670ef2f28d08d0",
    "original_candidate_line_sha256": "1a22c26d43540d9d102e0c10f71b03d39ecbb9544f1a213819670ef2f28d08d0",
    "mko_binding_revision": {
      "id": "edition-symbol-bindings-20261101",
      "path": "content/edition-symbol-bindings-20261101/revision.json",
      "sha256": "e9b4c69f2c4732074c37abb624c8ab8a70afc3645c12bf9df2268ef650ad7b6c",
      "previous_english_sha256": {
        "mko-euclidean-algorithm": "459ecc9798b00acbfa41527fab8e733798ec7e7b0713680d67ccb53a9a0cd5b8"
      }
    }
  },
  "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-20261026-008-p37",
    "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": "efee5462991619d6ed9f64f11ddb085e5614ef33f9f8fcd8cf29b9329f8fd301",
      "question_first_witness_sha256": "acf91fa1622f01d2cf04b2c513294254323665351f660f2628aaafdb869c4ff2"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 19,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "bd2d5adac9511899ec2b65db29222feac33c3a982b33951ca46c38ac3a9c6c3f",
      "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."
  }
}