{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261009-advance-20261013-r01-pythagorean-theorem-0036",
  "pilot_number": "R0011-06336",
  "number": 6336,
  "family_id": "qf-density13-pythagorean-hypotenuse-int",
  "title": {
    "zh-Hant": "第 6336 題：完全平方的正整數斜邊",
    "en": "Question 6336: Positive Integer Hypotenuse"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261009-advance-20261013-r01-pythagorean-theorem-0036",
    "batch_id": "batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-euclid-pythagorean-theorem"
    ],
    "title_zh": "完全平方的正整數斜邊",
    "stem_zh": "歐氏直角邊長模型；正股長，a=33、b=44，平方和已要求完全平方。求正整數斜邊c，不給負根。",
    "answer": {
      "kind": "expression",
      "value": "55",
      "explanation_zh": "正整數斜邊=55。用精確整數平方根核回平方和，再取正根；有限邊長檢查非新普遍證明，未完成任意仿射橋接。"
    },
    "difficulty": {
      "band": "upper_secondary",
      "level": 2,
      "skills": [
        "完全平方的正整數斜邊"
      ]
    },
    "variant": {
      "family_id": "qf-density13-pythagorean-hypotenuse-int",
      "seed": 202610131036,
      "parameters": {
        "a": 33,
        "b": 44,
        "geometry": "euclidean_right_side_model",
        "operation": "density13_pythagorean_hypotenuse_int",
        "domain": "positive_integer_right_side_protocol",
        "semantic_template": "qf-density13-pythagorean-hypotenuse-int/\"3025\""
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_integer_rational_arithmetic",
        "producer_finite_enumeration",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-09",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261009-advance-20261013-r01-pythagorean-theorem-0036",
    "batch_id": "batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "0b1ffdf2fdfb7169c18769876a72460445f2f25c8951ec0de20bacda86701015",
      "family_id": "qf-density13-pythagorean-hypotenuse-int",
      "seed": 202610131036,
      "semantic_template": "qf-density13-pythagorean-hypotenuse-int/\"3025\""
    },
    "fields": {
      "title": "Positive Integer Hypotenuse",
      "stem": "Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=33, b=44, with a perfect-square sum of squares. Find the positive integer hypotenuse c, not a negative root.",
      "answer": {
        "kind": "expression",
        "value": "55",
        "explanation": "Compute the exact integer principal square root and check its square; finite side checks are not new general formal Evidence. The exact result, retaining the requested order, is 55."
      },
      "skills": [
        "Positive Integer Hypotenuse"
      ],
      "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": "歐氏直角邊長模型；正股長，a=33、b=44，平方和已要求完全平方。求正整數斜邊c，不給負根。",
        "en": "Use a Euclidean right-triangle side-length model with strictly positive integer legs. a=33, b=44, with a perfect-square sum of squares. Find the positive integer hypotenuse c, not a negative root."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "Euclidean right-side model;positive leg a=33",
      "after": "use the supplied right-angle premise:a²+b²=c²",
      "relation": "implies",
      "rule_id": "pythagorean_right_side_premise",
      "conditions": [],
      "explanation": {
        "zh-Hant": "使用已宣告的歐氏直角邊長模型及正股長前提；此有限數值核對不是畢氏定理的新普遍證明，也不建立任意仿射幾何橋接。",
        "en": "Use the declared Euclidean right-side model with positive legs.This finite numerical check is not a new universal proof of the Pythagorean theorem or an arbitrary affine-geometric bridge."
      },
      "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": "known leg a squared := square(33)",
      "after": "known leg a squared = 1089",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先完整平方已知正股長a。",
        "en": "Square the complete known positive leg a."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 known leg a squared，不以小數近似替代。",
          "en": "Recompute known leg a squared with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "square",
        "inputs": [
          "33"
        ],
        "output": "1089",
        "label": "known leg a squared"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "leg b squared := square(44)",
      "after": "leg b squared = 1936",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "另一正股長b獨立平方。",
        "en": "Square the other positive leg b separately."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 leg b squared，不以小數近似替代。",
          "en": "Recompute leg b squared with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "square",
        "inputs": [
          "44"
        ],
        "output": "1936",
        "label": "leg b squared"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "sum of leg squares := sum(1089, 1936)",
      "after": "sum of leg squares = 3025",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "股平方相加是模型中的斜邊平方，不是斜邊本身。",
        "en": "The sum of leg squares is the model's squared hypotenuse,not the hypotenuse itself."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 sum of leg squares，不以小數近似替代。",
          "en": "Recompute sum of leg squares with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1089",
          "1936"
        ],
        "output": "3025",
        "label": "sum of leg squares"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "hypotenuse principal root squared := square(55)",
      "after": "hypotenuse principal root squared = 3025",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "用整數根平方反核正完全平方前提，不以浮點近似取整。",
        "en": "Square the integer root to audit the positive-perfect-square premise,without rounded floating roots."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 hypotenuse principal root squared，不以小數近似替代。",
          "en": "Recompute hypotenuse principal root squared with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "square",
        "inputs": [
          "55"
        ],
        "output": "3025",
        "label": "hypotenuse principal root squared"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "root=55>0;root²=3025=required square",
      "after": "unique positive length=55",
      "relation": "implies",
      "rule_id": "positive_integer_length_root",
      "conditions": [],
      "explanation": {
        "zh-Hant": "平方相等還須取正長度；負根不是邊長，零根也不符合本題非退化正股協議。",
        "en": "Matching the square still requires a positive length;a negative root is not a side length,and0 violates this nondegenerate positive-leg protocol."
      },
      "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": 55,
    "radicand": "3025"
  },
  "conclusion": {
    "canonical_value": "55",
    "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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      "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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    "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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}