{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261006-advance-20261007-r05-congruence-0063",
  "pilot_number": "R0005-03263",
  "number": 3263,
  "family_id": "qf-congruence-first-mismatch",
  "title": {
    "zh-Hant": "第 3263 題：首個不符的固定標記邊",
    "en": "Question 3263: First Mismatch in Fixed Corresponding Sides"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261006-advance-20261007-r05-congruence-0063",
    "batch_id": "batch-20261006-advance-20261007-r05-congruence-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-congruence"
    ],
    "title_zh": "首個不符的固定標記邊",
    "stem_zh": "等單位正交座標系，兩軸採同一長度單位。有序頂點 A=[[0,0],[13,0],[1,14]]、B=[[3,5],[16,5],[4,-9]]，兩者非退化且明示對應同索引。按 12、13、23 次序求首個平方長度不相等的邊標記，全相等答「無」。",
    "answer": {
      "kind": "short_text",
      "value": "無",
      "explanation_zh": "對應平方邊：A=[169, 197, 340]；B=[169, 197, 340]。「無」恰指本次三邊對應全相等；不偷偷重排頂點以消除不符。"
    },
    "difficulty": {
      "band": "lower_secondary",
      "level": 2,
      "skills": [
        "首個不符的固定標記邊"
      ]
    },
    "variant": {
      "family_id": "qf-congruence-first-mismatch",
      "seed": 202610075063,
      "parameters": {
        "A": [
          [
            0,
            0
          ],
          [
            13,
            0
          ],
          [
            1,
            14
          ]
        ],
        "B": [
          [
            3,
            5
          ],
          [
            16,
            5
          ],
          [
            4,
            -9
          ]
        ],
        "operation": "first_mismatch",
        "frame": "euclidean_orthonormal_equal_units",
        "semantic_template": "qf-congruence-first-mismatch/{\"inputs\":[[\"169\",\"197\",\"340\"],[\"169\",\"197\",\"340\"]],\"operation\":\"first_mismatch\"}"
      }
    },
    "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": "2026-10-06",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261006-advance-20261007-r05-congruence-0063",
    "batch_id": "batch-20261006-advance-20261007-r05-congruence-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "bb7346032c3585a50ecd8f257183109a847e93e1fd07e9321258da33fcc9d5eb",
      "family_id": "qf-congruence-first-mismatch",
      "seed": 202610075063,
      "semantic_template": "qf-congruence-first-mismatch/{\"inputs\":[[\"169\",\"197\",\"340\"],[\"169\",\"197\",\"340\"]],\"operation\":\"first_mismatch\"}"
    },
    "fields": {
      "title": "First Mismatch in Fixed Corresponding Sides",
      "stem": "Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(13, 0), A3=(1, 14) and B1=(3, 5), B2=(16, 5), B3=(4, -9), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. In order 12,13,23, give the first label whose corresponding squared side lengths differ. Answer none if all three agree.",
      "answer": {
        "kind": "short_text",
        "value": "none",
        "explanation": "The ordered squared sides are A=[169, 197, 340] and B=[169, 197, 340]. The first mismatch is none; none means agreement in this exact correspondence, without reordering vertices to remove a mismatch."
      },
      "skills": [
        "First Mismatch in Fixed Corresponding Sides"
      ],
      "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": "等單位正交座標系，兩軸採同一長度單位。有序頂點 A=[[0,0],[13,0],[1,14]]、B=[[3,5],[16,5],[4,-9]]，兩者非退化且明示對應同索引。按 12、13、23 次序求首個平方長度不相等的邊標記，全相等答「無」。",
        "en": "Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(13, 0), A3=(1, 14) and B1=(3, 5), B2=(16, 5), B3=(4, -9), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. In order 12,13,23, give the first label whose corresponding squared side lengths differ. Answer none if all three agree."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "ordered triples in an orthonormal equal-unit frame",
      "after": "vertex indices are fixed;allow rotations,translations and reflections",
      "relation": "implies",
      "rule_id": "fixed_vertex_correspondence",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按同索引對應，不自行重排頂點；歐氏全等允許反射。先檢查非退化，再用SSS。",
        "en": "Correspond equal indices without silently permuting vertices;Euclidean congruence allows reflection.Check nondegeneracy before SSS."
      },
      "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": "A vector1 component0 := sub(13, 0)",
      "after": "A vector1 component0 = 13",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "從第一頂點相減得到基向量分量。",
        "en": "Subtract the first vertex to obtain an edge-vector coordinate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 A vector1 component0，不以小數近似替代。",
          "en": "Recompute A vector1 component0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "13",
          "0"
        ],
        "output": "13",
        "label": "A vector1 component0"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "A vector1 component1 := sub(0, 0)",
      "after": "A vector1 component1 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "從第一頂點相減得到基向量分量。",
        "en": "Subtract the first vertex to obtain an edge-vector coordinate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 A vector1 component1，不以小數近似替代。",
          "en": "Recompute A vector1 component1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "0",
          "0"
        ],
        "output": "0",
        "label": "A vector1 component1"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "A vector2 component0 := sub(1, 0)",
      "after": "A vector2 component0 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "從第一頂點相減得到基向量分量。",
        "en": "Subtract the first vertex to obtain an edge-vector coordinate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 A vector2 component0，不以小數近似替代。",
          "en": "Recompute A vector2 component0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "1",
          "0"
        ],
        "output": "1",
        "label": "A vector2 component0"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "A vector2 component1 := sub(14, 0)",
      "after": "A vector2 component1 = 14",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "從第一頂點相減得到基向量分量。",
        "en": "Subtract the first vertex to obtain an edge-vector coordinate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 A vector2 component1，不以小數近似替代。",
          "en": "Recompute A vector2 component1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "14",
          "0"
        ],
        "output": "14",
        "label": "A vector2 component1"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "A determinant positive term := mul(13, 14)",
      "after": "A determinant positive term = 182",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "行列式先算第一個交叉乘積。",
        "en": "Compute the first cross product term of the determinant."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 A determinant positive term，不以小數近似替代。",
          "en": "Recompute A determinant positive term with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "13",
          "14"
        ],
        "output": "182",
        "label": "A determinant positive term"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "A determinant negative term := mul(0, 1)",
      "after": "A determinant negative term = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第二個交叉乘積保留減號位置。",
        "en": "Compute the other cross product term,retaining its subtraction position."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 A determinant negative term，不以小數近似替代。",
          "en": "Recompute A determinant negative term with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "1"
        ],
        "output": "0",
        "label": "A determinant negative term"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "A oriented double area := sub(182, 0)",
      "after": "A oriented double area = 182",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "兩項相減；非零當且僅當三點不共線，也排除重複頂點。",
        "en": "Subtract the terms;nonzero means noncollinear and also excludes repeated vertices."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 A oriented double area，不以小數近似替代。",
          "en": "Recompute A oriented double area with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "182",
          "0"
        ],
        "output": "182",
        "label": "A oriented double area"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "determinant A=182",
      "after": "proper triangle=是",
      "relation": "implies",
      "rule_id": "triangle_nondegeneracy",
      "conditions": [],
      "explanation": {
        "zh-Hant": "此精確行列式取代目測；退化三點不能被送入非退化三角形的SSS前提。",
        "en": "This exact determinant replaces visual guessing;degenerate triples do not meet the triangle premise of SSS."
      },
      "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": "B vector1 component0 := sub(16, 3)",
      "after": "B vector1 component0 = 13",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "從第一頂點相減得到基向量分量。",
        "en": "Subtract the first vertex to obtain an edge-vector coordinate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 B vector1 component0，不以小數近似替代。",
          "en": "Recompute B vector1 component0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "16",
          "3"
        ],
        "output": "13",
        "label": "B vector1 component0"
      }
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "B vector1 component1 := sub(5, 5)",
      "after": "B vector1 component1 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "從第一頂點相減得到基向量分量。",
        "en": "Subtract the first vertex to obtain an edge-vector coordinate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 B vector1 component1，不以小數近似替代。",
          "en": "Recompute B vector1 component1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "5",
          "5"
        ],
        "output": "0",
        "label": "B vector1 component1"
      }
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "B vector2 component0 := sub(4, 3)",
      "after": "B vector2 component0 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "從第一頂點相減得到基向量分量。",
        "en": "Subtract the first vertex to obtain an edge-vector coordinate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 B vector2 component0，不以小數近似替代。",
          "en": "Recompute B vector2 component0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "4",
          "3"
        ],
        "output": "1",
        "label": "B vector2 component0"
      }
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "B vector2 component1 := sub(-9, 5)",
      "after": "B vector2 component1 = -14",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "從第一頂點相減得到基向量分量。",
        "en": "Subtract the first vertex to obtain an edge-vector coordinate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 B vector2 component1，不以小數近似替代。",
          "en": "Recompute B vector2 component1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "-9",
          "5"
        ],
        "output": "-14",
        "label": "B vector2 component1"
      }
    },
    {
      "id": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "B determinant positive term := mul(13, -14)",
      "after": "B determinant positive term = -182",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "行列式先算第一個交叉乘積。",
        "en": "Compute the first cross product term of the determinant."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 B determinant positive term，不以小數近似替代。",
          "en": "Recompute B determinant positive term with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "13",
          "-14"
        ],
        "output": "-182",
        "label": "B determinant positive term"
      }
    },
    {
      "id": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "B determinant negative term := mul(0, 1)",
      "after": "B determinant negative term = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "第二個交叉乘積保留減號位置。",
        "en": "Compute the other cross product term,retaining its subtraction position."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 B determinant negative term，不以小數近似替代。",
          "en": "Recompute B determinant negative term with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "1"
        ],
        "output": "0",
        "label": "B determinant negative term"
      }
    },
    {
      "id": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "B oriented double area := sub(-182, 0)",
      "after": "B oriented double area = -182",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "兩項相減；非零當且僅當三點不共線，也排除重複頂點。",
        "en": "Subtract the terms;nonzero means noncollinear and also excludes repeated vertices."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 B oriented double area，不以小數近似替代。",
          "en": "Recompute B oriented double area with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "-182",
          "0"
        ],
        "output": "-182",
        "label": "B oriented double area"
      }
    },
    {
      "id": "S017",
      "uses": [
        "G001",
        "S016"
      ],
      "before": "determinant B=-182",
      "after": "proper triangle=是",
      "relation": "implies",
      "rule_id": "triangle_nondegeneracy",
      "conditions": [],
      "explanation": {
        "zh-Hant": "此精確行列式取代目測；退化三點不能被送入非退化三角形的SSS前提。",
        "en": "This exact determinant replaces visual guessing;degenerate triples do not meet the triangle premise of SSS."
      },
      "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": "S018",
      "uses": [
        "G001",
        "S017"
      ],
      "before": "A12 delta0 := sub(13, 0)",
      "after": "A12 delta0 = 13",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按同一邊的兩端座標求差。",
        "en": "Subtract coordinates of the endpoints of this same side."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 A12 delta0，不以小數近似替代。",
          "en": "Recompute A12 delta0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "13",
          "0"
        ],
        "output": "13",
        "label": "A12 delta0"
      }
    },
    {
      "id": "S019",
      "uses": [
        "G001",
        "S018"
      ],
      "before": "A12 square0 := square(13)",
      "after": "A12 square0 = 169",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐分量平方，正負差不影響長度。",
        "en": "Square the coordinate difference;its sign does not affect length."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 A12 square0，不以小數近似替代。",
          "en": "Recompute A12 square0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "square",
        "inputs": [
          "13"
        ],
        "output": "169",
        "label": "A12 square0"
      }
    },
    {
      "id": "S020",
      "uses": [
        "G001",
        "S019"
      ],
      "before": "A12 delta1 := sub(0, 0)",
      "after": "A12 delta1 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按同一邊的兩端座標求差。",
        "en": "Subtract coordinates of the endpoints of this same side."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 A12 delta1，不以小數近似替代。",
          "en": "Recompute A12 delta1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "0",
          "0"
        ],
        "output": "0",
        "label": "A12 delta1"
      }
    },
    {
      "id": "S021",
      "uses": [
        "G001",
        "S020"
      ],
      "before": "A12 square1 := square(0)",
      "after": "A12 square1 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐分量平方，正負差不影響長度。",
        "en": "Square the coordinate difference;its sign does not affect length."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 A12 square1，不以小數近似替代。",
          "en": "Recompute A12 square1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "square",
        "inputs": [
          "0"
        ],
        "output": "0",
        "label": "A12 square1"
      }
    },
    {
      "id": "S022",
      "uses": [
        "G001",
        "S021"
      ],
      "before": "A12 squared length := sum(169, 0)",
      "after": "A12 squared length = 169",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依畢氏關係相加，得到平方長度而不是邊長本身。",
        "en": "Add by the Pythagorean relation to obtain squared length,not length itself."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 A12 squared length，不以小數近似替代。",
          "en": "Recompute A12 squared length with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "169",
          "0"
        ],
        "output": "169",
        "label": "A12 squared length"
      }
    },
    {
      "id": "S023",
      "uses": [
        "G001",
        "S022"
      ],
      "before": "A13 delta0 := sub(1, 0)",
      "after": "A13 delta0 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按同一邊的兩端座標求差。",
        "en": "Subtract coordinates of the endpoints of this same side."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 A13 delta0，不以小數近似替代。",
          "en": "Recompute A13 delta0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "1",
          "0"
        ],
        "output": "1",
        "label": "A13 delta0"
      }
    },
    {
      "id": "S024",
      "uses": [
        "G001",
        "S023"
      ],
      "before": "A13 square0 := square(1)",
      "after": "A13 square0 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐分量平方，正負差不影響長度。",
        "en": "Square the coordinate difference;its sign does not affect length."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 A13 square0，不以小數近似替代。",
          "en": "Recompute A13 square0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "square",
        "inputs": [
          "1"
        ],
        "output": "1",
        "label": "A13 square0"
      }
    },
    {
      "id": "S025",
      "uses": [
        "G001",
        "S024"
      ],
      "before": "A13 delta1 := sub(14, 0)",
      "after": "A13 delta1 = 14",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按同一邊的兩端座標求差。",
        "en": "Subtract coordinates of the endpoints of this same side."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 A13 delta1，不以小數近似替代。",
          "en": "Recompute A13 delta1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "14",
          "0"
        ],
        "output": "14",
        "label": "A13 delta1"
      }
    },
    {
      "id": "S026",
      "uses": [
        "G001",
        "S025"
      ],
      "before": "A13 square1 := square(14)",
      "after": "A13 square1 = 196",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐分量平方，正負差不影響長度。",
        "en": "Square the coordinate difference;its sign does not affect length."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 A13 square1，不以小數近似替代。",
          "en": "Recompute A13 square1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "square",
        "inputs": [
          "14"
        ],
        "output": "196",
        "label": "A13 square1"
      }
    },
    {
      "id": "S027",
      "uses": [
        "G001",
        "S026"
      ],
      "before": "A13 squared length := sum(1, 196)",
      "after": "A13 squared length = 197",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依畢氏關係相加，得到平方長度而不是邊長本身。",
        "en": "Add by the Pythagorean relation to obtain squared length,not length itself."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 A13 squared length，不以小數近似替代。",
          "en": "Recompute A13 squared length with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1",
          "196"
        ],
        "output": "197",
        "label": "A13 squared length"
      }
    },
    {
      "id": "S028",
      "uses": [
        "G001",
        "S027"
      ],
      "before": "A23 delta0 := sub(1, 13)",
      "after": "A23 delta0 = -12",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按同一邊的兩端座標求差。",
        "en": "Subtract coordinates of the endpoints of this same side."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 A23 delta0，不以小數近似替代。",
          "en": "Recompute A23 delta0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "1",
          "13"
        ],
        "output": "-12",
        "label": "A23 delta0"
      }
    },
    {
      "id": "S029",
      "uses": [
        "G001",
        "S028"
      ],
      "before": "A23 square0 := square(-12)",
      "after": "A23 square0 = 144",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐分量平方，正負差不影響長度。",
        "en": "Square the coordinate difference;its sign does not affect length."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 A23 square0，不以小數近似替代。",
          "en": "Recompute A23 square0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "square",
        "inputs": [
          "-12"
        ],
        "output": "144",
        "label": "A23 square0"
      }
    },
    {
      "id": "S030",
      "uses": [
        "G001",
        "S029"
      ],
      "before": "A23 delta1 := sub(14, 0)",
      "after": "A23 delta1 = 14",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按同一邊的兩端座標求差。",
        "en": "Subtract coordinates of the endpoints of this same side."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 A23 delta1，不以小數近似替代。",
          "en": "Recompute A23 delta1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "14",
          "0"
        ],
        "output": "14",
        "label": "A23 delta1"
      }
    },
    {
      "id": "S031",
      "uses": [
        "G001",
        "S030"
      ],
      "before": "A23 square1 := square(14)",
      "after": "A23 square1 = 196",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐分量平方，正負差不影響長度。",
        "en": "Square the coordinate difference;its sign does not affect length."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 A23 square1，不以小數近似替代。",
          "en": "Recompute A23 square1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "square",
        "inputs": [
          "14"
        ],
        "output": "196",
        "label": "A23 square1"
      }
    },
    {
      "id": "S032",
      "uses": [
        "G001",
        "S031"
      ],
      "before": "A23 squared length := sum(144, 196)",
      "after": "A23 squared length = 340",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依畢氏關係相加，得到平方長度而不是邊長本身。",
        "en": "Add by the Pythagorean relation to obtain squared length,not length itself."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 A23 squared length，不以小數近似替代。",
          "en": "Recompute A23 squared length with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "144",
          "196"
        ],
        "output": "340",
        "label": "A23 squared length"
      }
    },
    {
      "id": "S033",
      "uses": [
        "G001",
        "S032"
      ],
      "before": "B12 delta0 := sub(16, 3)",
      "after": "B12 delta0 = 13",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按同一邊的兩端座標求差。",
        "en": "Subtract coordinates of the endpoints of this same side."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 B12 delta0，不以小數近似替代。",
          "en": "Recompute B12 delta0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "16",
          "3"
        ],
        "output": "13",
        "label": "B12 delta0"
      }
    },
    {
      "id": "S034",
      "uses": [
        "G001",
        "S033"
      ],
      "before": "B12 square0 := square(13)",
      "after": "B12 square0 = 169",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐分量平方，正負差不影響長度。",
        "en": "Square the coordinate difference;its sign does not affect length."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 B12 square0，不以小數近似替代。",
          "en": "Recompute B12 square0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "square",
        "inputs": [
          "13"
        ],
        "output": "169",
        "label": "B12 square0"
      }
    },
    {
      "id": "S035",
      "uses": [
        "G001",
        "S034"
      ],
      "before": "B12 delta1 := sub(5, 5)",
      "after": "B12 delta1 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按同一邊的兩端座標求差。",
        "en": "Subtract coordinates of the endpoints of this same side."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 B12 delta1，不以小數近似替代。",
          "en": "Recompute B12 delta1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "5",
          "5"
        ],
        "output": "0",
        "label": "B12 delta1"
      }
    },
    {
      "id": "S036",
      "uses": [
        "G001",
        "S035"
      ],
      "before": "B12 square1 := square(0)",
      "after": "B12 square1 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐分量平方，正負差不影響長度。",
        "en": "Square the coordinate difference;its sign does not affect length."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 B12 square1，不以小數近似替代。",
          "en": "Recompute B12 square1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "square",
        "inputs": [
          "0"
        ],
        "output": "0",
        "label": "B12 square1"
      }
    },
    {
      "id": "S037",
      "uses": [
        "G001",
        "S036"
      ],
      "before": "B12 squared length := sum(169, 0)",
      "after": "B12 squared length = 169",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依畢氏關係相加，得到平方長度而不是邊長本身。",
        "en": "Add by the Pythagorean relation to obtain squared length,not length itself."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 B12 squared length，不以小數近似替代。",
          "en": "Recompute B12 squared length with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "169",
          "0"
        ],
        "output": "169",
        "label": "B12 squared length"
      }
    },
    {
      "id": "S038",
      "uses": [
        "G001",
        "S037"
      ],
      "before": "B13 delta0 := sub(4, 3)",
      "after": "B13 delta0 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按同一邊的兩端座標求差。",
        "en": "Subtract coordinates of the endpoints of this same side."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 B13 delta0，不以小數近似替代。",
          "en": "Recompute B13 delta0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "4",
          "3"
        ],
        "output": "1",
        "label": "B13 delta0"
      }
    },
    {
      "id": "S039",
      "uses": [
        "G001",
        "S038"
      ],
      "before": "B13 square0 := square(1)",
      "after": "B13 square0 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐分量平方，正負差不影響長度。",
        "en": "Square the coordinate difference;its sign does not affect length."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 B13 square0，不以小數近似替代。",
          "en": "Recompute B13 square0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "square",
        "inputs": [
          "1"
        ],
        "output": "1",
        "label": "B13 square0"
      }
    },
    {
      "id": "S040",
      "uses": [
        "G001",
        "S039"
      ],
      "before": "B13 delta1 := sub(-9, 5)",
      "after": "B13 delta1 = -14",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按同一邊的兩端座標求差。",
        "en": "Subtract coordinates of the endpoints of this same side."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 B13 delta1，不以小數近似替代。",
          "en": "Recompute B13 delta1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "-9",
          "5"
        ],
        "output": "-14",
        "label": "B13 delta1"
      }
    },
    {
      "id": "S041",
      "uses": [
        "G001",
        "S040"
      ],
      "before": "B13 square1 := square(-14)",
      "after": "B13 square1 = 196",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐分量平方，正負差不影響長度。",
        "en": "Square the coordinate difference;its sign does not affect length."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 B13 square1，不以小數近似替代。",
          "en": "Recompute B13 square1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "square",
        "inputs": [
          "-14"
        ],
        "output": "196",
        "label": "B13 square1"
      }
    },
    {
      "id": "S042",
      "uses": [
        "G001",
        "S041"
      ],
      "before": "B13 squared length := sum(1, 196)",
      "after": "B13 squared length = 197",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依畢氏關係相加，得到平方長度而不是邊長本身。",
        "en": "Add by the Pythagorean relation to obtain squared length,not length itself."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 B13 squared length，不以小數近似替代。",
          "en": "Recompute B13 squared length with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1",
          "196"
        ],
        "output": "197",
        "label": "B13 squared length"
      }
    },
    {
      "id": "S043",
      "uses": [
        "G001",
        "S042"
      ],
      "before": "B23 delta0 := sub(4, 16)",
      "after": "B23 delta0 = -12",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按同一邊的兩端座標求差。",
        "en": "Subtract coordinates of the endpoints of this same side."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 B23 delta0，不以小數近似替代。",
          "en": "Recompute B23 delta0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "4",
          "16"
        ],
        "output": "-12",
        "label": "B23 delta0"
      }
    },
    {
      "id": "S044",
      "uses": [
        "G001",
        "S043"
      ],
      "before": "B23 square0 := square(-12)",
      "after": "B23 square0 = 144",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐分量平方，正負差不影響長度。",
        "en": "Square the coordinate difference;its sign does not affect length."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 B23 square0，不以小數近似替代。",
          "en": "Recompute B23 square0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "square",
        "inputs": [
          "-12"
        ],
        "output": "144",
        "label": "B23 square0"
      }
    },
    {
      "id": "S045",
      "uses": [
        "G001",
        "S044"
      ],
      "before": "B23 delta1 := sub(-9, 5)",
      "after": "B23 delta1 = -14",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按同一邊的兩端座標求差。",
        "en": "Subtract coordinates of the endpoints of this same side."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 B23 delta1，不以小數近似替代。",
          "en": "Recompute B23 delta1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "-9",
          "5"
        ],
        "output": "-14",
        "label": "B23 delta1"
      }
    },
    {
      "id": "S046",
      "uses": [
        "G001",
        "S045"
      ],
      "before": "B23 square1 := square(-14)",
      "after": "B23 square1 = 196",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐分量平方，正負差不影響長度。",
        "en": "Square the coordinate difference;its sign does not affect length."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 B23 square1，不以小數近似替代。",
          "en": "Recompute B23 square1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "square",
        "inputs": [
          "-14"
        ],
        "output": "196",
        "label": "B23 square1"
      }
    },
    {
      "id": "S047",
      "uses": [
        "G001",
        "S046"
      ],
      "before": "B23 squared length := sum(144, 196)",
      "after": "B23 squared length = 340",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依畢氏關係相加，得到平方長度而不是邊長本身。",
        "en": "Add by the Pythagorean relation to obtain squared length,not length itself."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 B23 squared length，不以小數近似替代。",
          "en": "Recompute B23 squared length with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "144",
          "196"
        ],
        "output": "340",
        "label": "B23 squared length"
      }
    },
    {
      "id": "S048",
      "uses": [
        "G001",
        "S047"
      ],
      "before": "side 12:A²=169;B²=169",
      "after": "corresponding lengths equal=是",
      "relation": "implies",
      "rule_id": "corresponding_side_equality",
      "conditions": [],
      "explanation": {
        "zh-Hant": "邊長非負，故平方相等當且僅當長度相等；只比較固定標記的同一條邊。",
        "en": "Lengths are nonnegative,so equality of squares is equivalent to equality of lengths.Compare only sides with the same fixed labels."
      },
      "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": "S049",
      "uses": [
        "G001",
        "S048"
      ],
      "before": "side 13:A²=197;B²=197",
      "after": "corresponding lengths equal=是",
      "relation": "implies",
      "rule_id": "corresponding_side_equality",
      "conditions": [],
      "explanation": {
        "zh-Hant": "邊長非負，故平方相等當且僅當長度相等；只比較固定標記的同一條邊。",
        "en": "Lengths are nonnegative,so equality of squares is equivalent to equality of lengths.Compare only sides with the same fixed labels."
      },
      "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": "S050",
      "uses": [
        "G001",
        "S049"
      ],
      "before": "side 23:A²=340;B²=340",
      "after": "corresponding lengths equal=是",
      "relation": "implies",
      "rule_id": "corresponding_side_equality",
      "conditions": [],
      "explanation": {
        "zh-Hant": "邊長非負，故平方相等當且僅當長度相等；只比較固定標記的同一條邊。",
        "en": "Lengths are nonnegative,so equality of squares is equivalent to equality of lengths.Compare only sides with the same fixed labels."
      },
      "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": "S051",
      "uses": [
        "G001",
        "S050"
      ],
      "before": "[是, 是, 是]",
      "after": "fixed-label SSS holds=是",
      "relation": "implies",
      "rule_id": "fixed_label_sss",
      "conditions": [],
      "explanation": {
        "zh-Hant": "非退化兩三角形的三條對應邊全相等才滿足SSS；三邊決定兩基向量的內積，故可由歐氏等距映射對應。任一不符只否定此標記，不否定其他排列。",
        "en": "SSS requires all three corresponding sides of the proper triangles to agree.The three sides determine the inner products of two edge vectors,allowing a Euclidean isometry.Any mismatch rejects this labeling,not every other permutation."
      },
      "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": "無",
    "determinants": [
      "182",
      "-182"
    ],
    "squared_sides": [
      [
        "169",
        "197",
        "340"
      ],
      [
        "169",
        "197",
        "340"
      ]
    ]
  },
  "conclusion": {
    "canonical_value": "無",
    "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-20261006-advance-20261007-r05-congruence-foundations",
      "batch_path": "public/data/questions/batches/batch-20261006-advance-20261007-r05-congruence-foundations/questions.jsonl",
      "batch_sha256": "72ed0f0e7041217534b522705a5a6ca20453393ddf56df1e9fbc5127963aa65d",
      "original_line": 63,
      "original_line_sha256": "896daafd4873a877519248715770794cd679f0fb76c33f32af0cad178d25bb55",
      "question_compact_sha256": "bb7346032c3585a50ecd8f257183109a847e93e1fd07e9321258da33fcc9d5eb",
      "english_row_compact_sha256": "c56e3368b381197f1f001fd28a08a615d3da943b1e06b557dab14be7fdce472e",
      "english_manifest_path": "public/data/i18n/editions/batch-20261006-advance-20261007-r05-congruence-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261006-advance-20261007-r05-congruence-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-congruence": {
        "sha256": "493629f3decdf708c53ca64d9ba2a4332bee59a979caa6d4767e0577fac7fbec",
        "english_sha256": "e83086a05f66a3d37e6976814fd8902f9c3dcf2d8ea8312c74f38348425f3457",
        "original_path": "public/data/mko/congruence.json",
        "english_path": "public/data/i18n/editions/mko-congruence.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "66b95c054a23815e6aa9999f9605667abcb96e0c5a97b71cd67b67041065878a",
    "original_candidate_sha256": "e0040a1aa7f40cce2d787c74f196158d99b1166047901b5d55f39b9d8a384488",
    "original_candidate_line_sha256": "e0040a1aa7f40cce2d787c74f196158d99b1166047901b5d55f39b9d8a384488"
  },
  "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-20261026-004-p6",
    "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": "5d74de11610ebc6037a4fc9946d77154d4ed18389f8f0feeb823148b6320c4c2",
      "question_first_witness_sha256": "1fba4ab0f13c2cfcc4d9fdebe492c442d43dd45700394a443daa28f333355ed9"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 51,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "fd08c94263c673953ef3afaae7876def198b544d2dbeb380661a67bf5de191bd",
      "arithmetic_steps": 44
    },
    "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."
  }
}