{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261009-advance-20261013-r01-pythagorean-theorem-0061",
  "pilot_number": "R0011-06386",
  "number": 6386,
  "family_id": "qf-density13-pythagorean-missing-leg-int",
  "title": {
    "zh-Hant": "第 6386 題：正整數缺股",
    "en": "Question 6386: Positive Integer Missing Leg"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261009-advance-20261013-r01-pythagorean-theorem-0061",
    "batch_id": "batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-euclid-pythagorean-theorem"
    ],
    "title_zh": "正整數缺股",
    "stem_zh": "歐氏直角邊長模型；正股長，已知a=33、正斜邊c=55且c>a，平方差為正完全平方。求另一正股b。",
    "answer": {
      "kind": "expression",
      "value": "44",
      "explanation_zh": "正整數缺股=44。先扣已知股平方，再核正整數主根；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": [
        "正整數缺股"
      ]
    },
    "variant": {
      "family_id": "qf-density13-pythagorean-missing-leg-int",
      "seed": 202610131061,
      "parameters": {
        "a": 33,
        "c": 55,
        "geometry": "euclidean_right_side_model",
        "operation": "density13_pythagorean_missing_leg_int",
        "domain": "positive_integer_right_side_protocol",
        "semantic_template": "qf-density13-pythagorean-missing-leg-int/\"1936\""
      }
    },
    "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-0061",
    "batch_id": "batch-20261009-advance-20261013-r01-pythagorean-theorem-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "b8977d987826df7bce7c8a6c7704e09e964bb3f8f00ae4533e0944f28fe86b01",
      "family_id": "qf-density13-pythagorean-missing-leg-int",
      "seed": 202610131061,
      "semantic_template": "qf-density13-pythagorean-missing-leg-int/\"1936\""
    },
    "fields": {
      "title": "Positive Integer Missing Leg",
      "stem": "Use a Euclidean right-triangle side-length model with strictly positive integer legs. Known leg a=33, positive hypotenuse c=55>a; c²−a² is a positive perfect square. Find the other positive leg b.",
      "answer": {
        "kind": "expression",
        "value": "44",
        "explanation": "Subtract the known leg square before taking the positive exact integer root; do not claim a general geometry bridge. The exact result, retaining the requested order, is 44."
      },
      "skills": [
        "Positive Integer Missing Leg"
      ],
      "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，已知a=33、正斜邊c=55且c>a，平方差為正完全平方。求另一正股b。",
        "en": "Use a Euclidean right-triangle side-length model with strictly positive integer legs. Known leg a=33, positive hypotenuse c=55>a; c²−a² is a positive perfect square. Find the other positive leg b."
      }
    }
  ],
  "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": "hypotenuse c squared := square(55)",
      "after": "hypotenuse c squared = 3025",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正斜邊c先平方。",
        "en": "Square the positive hypotenuse c."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 hypotenuse c squared，不以小數近似替代。",
          "en": "Recompute hypotenuse c squared with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "square",
        "inputs": [
          "55"
        ],
        "output": "3025",
        "label": "hypotenuse c squared"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "missing leg squared := sub(3025, 1089)",
      "after": "missing leg squared = 1936",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "由b²=c²−a²扣已知股平方，必須為正完全平方。",
        "en": "Rearrange b²=c²−a² by subtracting the known leg square;the result must be a positive perfect square."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 missing leg squared，不以小數近似替代。",
          "en": "Recompute missing leg squared with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "3025",
          "1089"
        ],
        "output": "1936",
        "label": "missing leg squared"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "missing leg principal root squared := square(44)",
      "after": "missing leg principal root squared = 1936",
      "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 missing leg principal root squared，不以小數近似替代。",
          "en": "Recompute missing leg principal root squared with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "square",
        "inputs": [
          "44"
        ],
        "output": "1936",
        "label": "missing leg principal root squared"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "root=44>0;root²=1936=required square",
      "after": "unique positive length=44",
      "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": 44,
    "radicand": "1936"
  },
  "conclusion": {
    "canonical_value": "44",
    "explanation": {
      "zh-Hant": "依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。",
      "en": "The full substitutions and premise checks give the original answer; source identity,format and status are unchanged."
    }
  },
  "definition_note": {
    "zh-Hant": "沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。",
    "en": "Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate."
  },
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    "original_candidate_line_sha256": "951806242995854f0ae21ddc2fdbdadb81ccc8b13a710f9594eaf903da067141"
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  "historical_source_state": {
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      "source_question_state": "candidate",
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      "formalization": "not_formalized",
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      "zh-Hant": "來源綁定教學推導候選；精確核對不等於獨立AI／人工審查或一般形式證明。原題、答案、MKO、公式、AST與歷史不變。",
      "en": "Source-bound pedagogical derivation candidate.Exact checking is not independent AI/human review or general formal proof.Original questions,answers,MKOs,formulas,ASTs and history remain unchanged."
    },
    "label": "Frozen producer metadata; current per-question review is separate below."
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  "revision": {
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    "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.",
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    "bilingual_semantics": true,
    "steps_reviewed": 6,
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    "Lean": "NOT_RUN",
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    "authority": "Owner-authorized reviewed reader publication; original question/candidate states remain historical and unchanged."
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}