{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261006-advance-20261007-r06-similarity-0011",
  "pilot_number": "R0056-28886",
  "number": 28886,
  "family_id": "qf-similarity-specified-scale",
  "title": {
    "zh-Hant": "第 28886 題：指定正倍率的標記相似見證",
    "en": "Question 28886: Labeled Similarity at a Specified Positive Scale"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261006-advance-20261007-r06-similarity-0011",
    "batch_id": "batch-20261006-advance-20261007-r06-similarity-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-similarity"
    ],
    "title_zh": "指定正倍率的標記相似見證",
    "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],[3,0],[1,2]]、B=[[3,-4],[36,-4],[14,-26]]，兩者非退化且明示對應同索引；指定正長度倍率 k=B/A=11/1。這個倍率是否符合全部三條對應邊？",
    "answer": {
      "kind": "short_text",
      "value": "是",
      "explanation_zh": "指定倍率=11/1；A平方邊=[9, 5, 8]；B平方邊=[1089, 605, 968]。以平方邊長比較 B²q²=A²p²；本次見證的否定不是其他倍率的否定，也不變更標記對應。"
    },
    "difficulty": {
      "band": "lower_secondary",
      "level": 2,
      "skills": [
        "指定正倍率的標記相似見證"
      ]
    },
    "variant": {
      "family_id": "qf-similarity-specified-scale",
      "seed": 202610076011,
      "parameters": {
        "A": [
          [
            0,
            0
          ],
          [
            3,
            0
          ],
          [
            1,
            2
          ]
        ],
        "B": [
          [
            3,
            -4
          ],
          [
            36,
            -4
          ],
          [
            14,
            -26
          ]
        ],
        "p": 11,
        "q": 1,
        "operation": "specified_similarity",
        "frame": "euclidean_orthonormal_equal_units",
        "semantic_template": "qf-similarity-specified-scale/{\"inputs\":[[\"121/1\",\"121/1\",\"121/1\"],\"11/1\"],\"operation\":\"specified_similarity\"}"
      }
    },
    "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-r06-similarity-0011",
    "batch_id": "batch-20261006-advance-20261007-r06-similarity-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "aa0d04f7f11cea00a3d91d02f2b6f2c11db37dbd4cb6d2a07eebc9a50670bce5",
      "family_id": "qf-similarity-specified-scale",
      "seed": 202610076011,
      "semantic_template": "qf-similarity-specified-scale/{\"inputs\":[[\"121/1\",\"121/1\",\"121/1\"],\"11/1\"],\"operation\":\"specified_similarity\"}"
    },
    "fields": {
      "title": "Labeled Similarity at a Specified Positive Scale",
      "stem": "Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(3, 0), A3=(1, 2) and B1=(3, -4), B2=(36, -4), B3=(14, -26), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. The specified positive length scale from A to B is k=11/1. Does this scale fit every corresponding side?",
      "answer": {
        "kind": "short_text",
        "value": "Yes",
        "explanation": "The squared sides are A=[9, 5, 8] and B=[1089, 605, 968]. Check B²q²=A²p² on each corresponding side, obtaining Yes. Rejecting this specified scale witness does not reject another possible scale or change the vertex correspondence."
      },
      "skills": [
        "Labeled Similarity at a Specified Positive Scale"
      ],
      "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],[3,0],[1,2]]、B=[[3,-4],[36,-4],[14,-26]]，兩者非退化且明示對應同索引；指定正長度倍率 k=B/A=11/1。這個倍率是否符合全部三條對應邊？",
        "en": "Use one equal-unit orthonormal Euclidean Cartesian frame. The ordered nondegenerate triangles have A1=(0, 0), A2=(3, 0), A3=(1, 2) and B1=(3, -4), B2=(36, -4), B3=(14, -26), with explicit correspondence A1↔B1, A2↔B2, A3↔B3. The specified positive length scale from A to B is k=11/1. Does this scale fit every corresponding side?"
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source roles = [qf-similarity-specified-scale, specified_similarity]",
      "after": "derived = [ALL_fixed_corresponding_sides_at_given_forward_scale]",
      "relation": "implies",
      "rule_id": "R56_source_answer_role_and_EXACT_family_interface",
      "conditions": [],
      "explanation": {
        "zh-Hant": "本題輸出依原scalar／set／是非／fraction角色；有理數整數保留/1。不加未宣告domain，不改中英來源、公式或題目。",
        "en": "Use the declared scalar/set/Yes-No/fraction answer role;integer fractions retain/1.Do not add an undeclared domain or change original bilingual data,formulas or questions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向逐項代回原始資料與本步條件；保持索引、符號、嚴格／非嚴格邊界，不能用正確答案代替推導責任。這是候選教學與精確核對，非形式證明或發布資格。",
          "en": "Reverse-check EACH original datum and premise,retaining indices,signs and strict/non-strict boundaries.A correct final answer cannot replace the derivation obligations.This is candidate teaching/exact checking,not formal proof or publication clearance."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "Similarity ORIGINAL scale numerator squared := mul(11, 11)",
      "after": "Similarity ORIGINAL scale numerator squared = 121",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本題原始角色作精確有理運算；SAFE只約束原輸入，衍生整數不截斷，不作浮點近似。",
        "en": "Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Similarity ORIGINAL scale numerator squared，不以小數近似替代。",
          "en": "Recompute Similarity ORIGINAL scale numerator squared with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "11",
          "11"
        ],
        "output": "121",
        "label": "Similarity ORIGINAL scale numerator squared"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "Similarity ORIGINAL scale denominator squared := mul(1, 1)",
      "after": "Similarity ORIGINAL scale denominator squared = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本題原始角色作精確有理運算；SAFE只約束原輸入，衍生整數不截斷，不作浮點近似。",
        "en": "Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Similarity ORIGINAL scale denominator squared，不以小數近似替代。",
          "en": "Recompute Similarity ORIGINAL scale denominator squared with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "1"
        ],
        "output": "1",
        "label": "Similarity ORIGINAL scale denominator squared"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "Similarity A determinant ORIGINAL point1 coordinate0 := sub(3, 0)",
      "after": "Similarity A determinant ORIGINAL point1 coordinate0 = 3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本題原始角色作精確有理運算；SAFE只約束原輸入，衍生整數不截斷，不作浮點近似。",
        "en": "Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Similarity A determinant ORIGINAL point1 coordinate0，不以小數近似替代。",
          "en": "Recompute Similarity A determinant ORIGINAL point1 coordinate0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "3",
          "0"
        ],
        "output": "3",
        "label": "Similarity A determinant ORIGINAL point1 coordinate0"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "Similarity A determinant ORIGINAL point1 coordinate1 := sub(0, 0)",
      "after": "Similarity A determinant ORIGINAL point1 coordinate1 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本題原始角色作精確有理運算；SAFE只約束原輸入，衍生整數不截斷，不作浮點近似。",
        "en": "Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Similarity A determinant ORIGINAL point1 coordinate1，不以小數近似替代。",
          "en": "Recompute Similarity A determinant ORIGINAL point1 coordinate1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "0",
          "0"
        ],
        "output": "0",
        "label": "Similarity A determinant ORIGINAL point1 coordinate1"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "Similarity A determinant ORIGINAL point2 coordinate0 := sub(1, 0)",
      "after": "Similarity A determinant ORIGINAL point2 coordinate0 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本題原始角色作精確有理運算；SAFE只約束原輸入，衍生整數不截斷，不作浮點近似。",
        "en": "Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Similarity A determinant ORIGINAL point2 coordinate0，不以小數近似替代。",
          "en": "Recompute Similarity A determinant ORIGINAL point2 coordinate0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "1",
          "0"
        ],
        "output": "1",
        "label": "Similarity A determinant ORIGINAL point2 coordinate0"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "Similarity A determinant ORIGINAL point2 coordinate1 := sub(2, 0)",
      "after": "Similarity A determinant ORIGINAL point2 coordinate1 = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本題原始角色作精確有理運算；SAFE只約束原輸入，衍生整數不截斷，不作浮點近似。",
        "en": "Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Similarity A determinant ORIGINAL point2 coordinate1，不以小數近似替代。",
          "en": "Recompute Similarity A determinant ORIGINAL point2 coordinate1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "2",
          "0"
        ],
        "output": "2",
        "label": "Similarity A determinant ORIGINAL point2 coordinate1"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "Similarity A determinant ORIGINAL first cross product := mul(3, 2)",
      "after": "Similarity A determinant ORIGINAL first cross product = 6",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本題原始角色作精確有理運算；SAFE只約束原輸入，衍生整數不截斷，不作浮點近似。",
        "en": "Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Similarity A determinant ORIGINAL first cross product，不以小數近似替代。",
          "en": "Recompute Similarity A determinant ORIGINAL first cross product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "3",
          "2"
        ],
        "output": "6",
        "label": "Similarity A determinant ORIGINAL first cross product"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "Similarity A determinant ORIGINAL second cross product := mul(0, 1)",
      "after": "Similarity A determinant ORIGINAL second cross product = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本題原始角色作精確有理運算；SAFE只約束原輸入，衍生整數不截斷，不作浮點近似。",
        "en": "Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Similarity A determinant ORIGINAL second cross product，不以小數近似替代。",
          "en": "Recompute Similarity A determinant ORIGINAL second cross product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "1"
        ],
        "output": "0",
        "label": "Similarity A determinant ORIGINAL second cross product"
      }
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "Similarity A determinant ORIGINAL cross difference := sub(6, 0)",
      "after": "Similarity A determinant ORIGINAL cross difference = 6",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本題原始角色作精確有理運算；SAFE只約束原輸入，衍生整數不截斷，不作浮點近似。",
        "en": "Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Similarity A determinant ORIGINAL cross difference，不以小數近似替代。",
          "en": "Recompute Similarity A determinant ORIGINAL cross difference with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "6",
          "0"
        ],
        "output": "6",
        "label": "Similarity A determinant ORIGINAL cross difference"
      }
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "source roles = [Similarity A, [[0, 0], [3, 0], [1, 2]]]",
      "after": "derived = [6, 是]",
      "relation": "implies",
      "rule_id": "Similarity_ACTUAL_proper_ordered_triangle_no_relabel",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先驗兩邊各為真nondegenerate triangle：兩條基點向量cross不等0。保原A1,A2,A3或B1,B2,B3次序，不能重排找另一個配對。",
        "en": "Require an ACTUAL nondegenerate triangle via the nonzero cross product of two basepoint vectors.Retain original A1,A2,A3 or B1,B2,B3 labels;do not search another correspondence."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向逐項代回原始資料與本步條件；保持索引、符號、嚴格／非嚴格邊界，不能用正確答案代替推導責任。這是候選教學與精確核對，非形式證明或發布資格。",
          "en": "Reverse-check EACH original datum and premise,retaining indices,signs and strict/non-strict boundaries.A correct final answer cannot replace the derivation obligations.This is candidate teaching/exact checking,not formal proof or publication clearance."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "Similarity A side12 ORIGINAL coordinate0 difference := sub(3, 0)",
      "after": "Similarity A side12 ORIGINAL coordinate0 difference = 3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本題原始角色作精確有理運算；SAFE只約束原輸入，衍生整數不截斷，不作浮點近似。",
        "en": "Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Similarity A side12 ORIGINAL coordinate0 difference，不以小數近似替代。",
          "en": "Recompute Similarity A side12 ORIGINAL coordinate0 difference with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "3",
          "0"
        ],
        "output": "3",
        "label": "Similarity A side12 ORIGINAL coordinate0 difference"
      }
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "Similarity A side12 ORIGINAL coordinate1 difference := sub(0, 0)",
      "after": "Similarity A side12 ORIGINAL coordinate1 difference = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本題原始角色作精確有理運算；SAFE只約束原輸入，衍生整數不截斷，不作浮點近似。",
        "en": "Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Similarity A side12 ORIGINAL coordinate1 difference，不以小數近似替代。",
          "en": "Recompute Similarity A side12 ORIGINAL coordinate1 difference with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "0",
          "0"
        ],
        "output": "0",
        "label": "Similarity A side12 ORIGINAL coordinate1 difference"
      }
    },
    {
      "id": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "Similarity A side12 ORIGINAL coordinate0 square := mul(3, 3)",
      "after": "Similarity A side12 ORIGINAL coordinate0 square = 9",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本題原始角色作精確有理運算；SAFE只約束原輸入，衍生整數不截斷，不作浮點近似。",
        "en": "Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Similarity A side12 ORIGINAL coordinate0 square，不以小數近似替代。",
          "en": "Recompute Similarity A side12 ORIGINAL coordinate0 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "3",
          "3"
        ],
        "output": "9",
        "label": "Similarity A side12 ORIGINAL coordinate0 square"
      }
    },
    {
      "id": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "Similarity A side12 ORIGINAL coordinate1 square := mul(0, 0)",
      "after": "Similarity A side12 ORIGINAL coordinate1 square = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本題原始角色作精確有理運算；SAFE只約束原輸入，衍生整數不截斷，不作浮點近似。",
        "en": "Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Similarity A side12 ORIGINAL coordinate1 square，不以小數近似替代。",
          "en": "Recompute Similarity A side12 ORIGINAL coordinate1 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "0"
        ],
        "output": "0",
        "label": "Similarity A side12 ORIGINAL coordinate1 square"
      }
    },
    {
      "id": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "Similarity A side12 ORIGINAL squared length := sum(9, 0)",
      "after": "Similarity A side12 ORIGINAL squared length = 9",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本題原始角色作精確有理運算；SAFE只約束原輸入，衍生整數不截斷，不作浮點近似。",
        "en": "Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Similarity A side12 ORIGINAL squared length，不以小數近似替代。",
          "en": "Recompute Similarity A side12 ORIGINAL squared length with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "9",
          "0"
        ],
        "output": "9",
        "label": "Similarity A side12 ORIGINAL squared length"
      }
    },
    {
      "id": "S017",
      "uses": [
        "G001",
        "S016"
      ],
      "before": "Similarity A side13 ORIGINAL coordinate0 difference := sub(1, 0)",
      "after": "Similarity A side13 ORIGINAL coordinate0 difference = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本題原始角色作精確有理運算；SAFE只約束原輸入，衍生整數不截斷，不作浮點近似。",
        "en": "Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Similarity A side13 ORIGINAL coordinate0 difference，不以小數近似替代。",
          "en": "Recompute Similarity A side13 ORIGINAL coordinate0 difference with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "1",
          "0"
        ],
        "output": "1",
        "label": "Similarity A side13 ORIGINAL coordinate0 difference"
      }
    },
    {
      "id": "S018",
      "uses": [
        "G001",
        "S017"
      ],
      "before": "Similarity A side13 ORIGINAL coordinate1 difference := sub(2, 0)",
      "after": "Similarity A side13 ORIGINAL coordinate1 difference = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本題原始角色作精確有理運算；SAFE只約束原輸入，衍生整數不截斷，不作浮點近似。",
        "en": "Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Similarity A side13 ORIGINAL coordinate1 difference，不以小數近似替代。",
          "en": "Recompute Similarity A side13 ORIGINAL coordinate1 difference with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "2",
          "0"
        ],
        "output": "2",
        "label": "Similarity A side13 ORIGINAL coordinate1 difference"
      }
    },
    {
      "id": "S019",
      "uses": [
        "G001",
        "S018"
      ],
      "before": "Similarity A side13 ORIGINAL coordinate0 square := mul(1, 1)",
      "after": "Similarity A side13 ORIGINAL coordinate0 square = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本題原始角色作精確有理運算；SAFE只約束原輸入，衍生整數不截斷，不作浮點近似。",
        "en": "Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Similarity A side13 ORIGINAL coordinate0 square，不以小數近似替代。",
          "en": "Recompute Similarity A side13 ORIGINAL coordinate0 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "1"
        ],
        "output": "1",
        "label": "Similarity A side13 ORIGINAL coordinate0 square"
      }
    },
    {
      "id": "S020",
      "uses": [
        "G001",
        "S019"
      ],
      "before": "Similarity A side13 ORIGINAL coordinate1 square := mul(2, 2)",
      "after": "Similarity A side13 ORIGINAL coordinate1 square = 4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本題原始角色作精確有理運算；SAFE只約束原輸入，衍生整數不截斷，不作浮點近似。",
        "en": "Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Similarity A side13 ORIGINAL coordinate1 square，不以小數近似替代。",
          "en": "Recompute Similarity A side13 ORIGINAL coordinate1 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "2"
        ],
        "output": "4",
        "label": "Similarity A side13 ORIGINAL coordinate1 square"
      }
    },
    {
      "id": "S021",
      "uses": [
        "G001",
        "S020"
      ],
      "before": "Similarity A side13 ORIGINAL squared length := sum(1, 4)",
      "after": "Similarity A side13 ORIGINAL squared length = 5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本題原始角色作精確有理運算；SAFE只約束原輸入，衍生整數不截斷，不作浮點近似。",
        "en": "Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Similarity A side13 ORIGINAL squared length，不以小數近似替代。",
          "en": "Recompute Similarity A side13 ORIGINAL squared length with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1",
          "4"
        ],
        "output": "5",
        "label": "Similarity A side13 ORIGINAL squared length"
      }
    },
    {
      "id": "S022",
      "uses": [
        "G001",
        "S021"
      ],
      "before": "Similarity A side23 ORIGINAL coordinate0 difference := sub(1, 3)",
      "after": "Similarity A side23 ORIGINAL coordinate0 difference = -2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本題原始角色作精確有理運算；SAFE只約束原輸入，衍生整數不截斷，不作浮點近似。",
        "en": "Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Similarity A side23 ORIGINAL coordinate0 difference，不以小數近似替代。",
          "en": "Recompute Similarity A side23 ORIGINAL coordinate0 difference with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "1",
          "3"
        ],
        "output": "-2",
        "label": "Similarity A side23 ORIGINAL coordinate0 difference"
      }
    },
    {
      "id": "S023",
      "uses": [
        "G001",
        "S022"
      ],
      "before": "Similarity A side23 ORIGINAL coordinate1 difference := sub(2, 0)",
      "after": "Similarity A side23 ORIGINAL coordinate1 difference = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本題原始角色作精確有理運算；SAFE只約束原輸入，衍生整數不截斷，不作浮點近似。",
        "en": "Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Similarity A side23 ORIGINAL coordinate1 difference，不以小數近似替代。",
          "en": "Recompute Similarity A side23 ORIGINAL coordinate1 difference with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "2",
          "0"
        ],
        "output": "2",
        "label": "Similarity A side23 ORIGINAL coordinate1 difference"
      }
    },
    {
      "id": "S024",
      "uses": [
        "G001",
        "S023"
      ],
      "before": "Similarity A side23 ORIGINAL coordinate0 square := mul(-2, -2)",
      "after": "Similarity A side23 ORIGINAL coordinate0 square = 4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本題原始角色作精確有理運算；SAFE只約束原輸入，衍生整數不截斷，不作浮點近似。",
        "en": "Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Similarity A side23 ORIGINAL coordinate0 square，不以小數近似替代。",
          "en": "Recompute Similarity A side23 ORIGINAL coordinate0 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-2",
          "-2"
        ],
        "output": "4",
        "label": "Similarity A side23 ORIGINAL coordinate0 square"
      }
    },
    {
      "id": "S025",
      "uses": [
        "G001",
        "S024"
      ],
      "before": "Similarity A side23 ORIGINAL coordinate1 square := mul(2, 2)",
      "after": "Similarity A side23 ORIGINAL coordinate1 square = 4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本題原始角色作精確有理運算；SAFE只約束原輸入，衍生整數不截斷，不作浮點近似。",
        "en": "Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Similarity A side23 ORIGINAL coordinate1 square，不以小數近似替代。",
          "en": "Recompute Similarity A side23 ORIGINAL coordinate1 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "2"
        ],
        "output": "4",
        "label": "Similarity A side23 ORIGINAL coordinate1 square"
      }
    },
    {
      "id": "S026",
      "uses": [
        "G001",
        "S025"
      ],
      "before": "Similarity A side23 ORIGINAL squared length := sum(4, 4)",
      "after": "Similarity A side23 ORIGINAL squared length = 8",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本題原始角色作精確有理運算；SAFE只約束原輸入，衍生整數不截斷，不作浮點近似。",
        "en": "Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Similarity A side23 ORIGINAL squared length，不以小數近似替代。",
          "en": "Recompute Similarity A side23 ORIGINAL squared length with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "4",
          "4"
        ],
        "output": "8",
        "label": "Similarity A side23 ORIGINAL squared length"
      }
    },
    {
      "id": "S027",
      "uses": [
        "G001",
        "S026"
      ],
      "before": "Similarity B determinant ORIGINAL point1 coordinate0 := sub(36, 3)",
      "after": "Similarity B determinant ORIGINAL point1 coordinate0 = 33",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本題原始角色作精確有理運算；SAFE只約束原輸入，衍生整數不截斷，不作浮點近似。",
        "en": "Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Similarity B determinant ORIGINAL point1 coordinate0，不以小數近似替代。",
          "en": "Recompute Similarity B determinant ORIGINAL point1 coordinate0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "36",
          "3"
        ],
        "output": "33",
        "label": "Similarity B determinant ORIGINAL point1 coordinate0"
      }
    },
    {
      "id": "S028",
      "uses": [
        "G001",
        "S027"
      ],
      "before": "Similarity B determinant ORIGINAL point1 coordinate1 := sub(-4, -4)",
      "after": "Similarity B determinant ORIGINAL point1 coordinate1 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本題原始角色作精確有理運算；SAFE只約束原輸入，衍生整數不截斷，不作浮點近似。",
        "en": "Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Similarity B determinant ORIGINAL point1 coordinate1，不以小數近似替代。",
          "en": "Recompute Similarity B determinant ORIGINAL point1 coordinate1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "-4",
          "-4"
        ],
        "output": "0",
        "label": "Similarity B determinant ORIGINAL point1 coordinate1"
      }
    },
    {
      "id": "S029",
      "uses": [
        "G001",
        "S028"
      ],
      "before": "Similarity B determinant ORIGINAL point2 coordinate0 := sub(14, 3)",
      "after": "Similarity B determinant ORIGINAL point2 coordinate0 = 11",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本題原始角色作精確有理運算；SAFE只約束原輸入，衍生整數不截斷，不作浮點近似。",
        "en": "Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Similarity B determinant ORIGINAL point2 coordinate0，不以小數近似替代。",
          "en": "Recompute Similarity B determinant ORIGINAL point2 coordinate0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "14",
          "3"
        ],
        "output": "11",
        "label": "Similarity B determinant ORIGINAL point2 coordinate0"
      }
    },
    {
      "id": "S030",
      "uses": [
        "G001",
        "S029"
      ],
      "before": "Similarity B determinant ORIGINAL point2 coordinate1 := sub(-26, -4)",
      "after": "Similarity B determinant ORIGINAL point2 coordinate1 = -22",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本題原始角色作精確有理運算；SAFE只約束原輸入，衍生整數不截斷，不作浮點近似。",
        "en": "Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Similarity B determinant ORIGINAL point2 coordinate1，不以小數近似替代。",
          "en": "Recompute Similarity B determinant ORIGINAL point2 coordinate1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "-26",
          "-4"
        ],
        "output": "-22",
        "label": "Similarity B determinant ORIGINAL point2 coordinate1"
      }
    },
    {
      "id": "S031",
      "uses": [
        "G001",
        "S030"
      ],
      "before": "Similarity B determinant ORIGINAL first cross product := mul(33, -22)",
      "after": "Similarity B determinant ORIGINAL first cross product = -726",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本題原始角色作精確有理運算；SAFE只約束原輸入，衍生整數不截斷，不作浮點近似。",
        "en": "Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Similarity B determinant ORIGINAL first cross product，不以小數近似替代。",
          "en": "Recompute Similarity B determinant ORIGINAL first cross product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "33",
          "-22"
        ],
        "output": "-726",
        "label": "Similarity B determinant ORIGINAL first cross product"
      }
    },
    {
      "id": "S032",
      "uses": [
        "G001",
        "S031"
      ],
      "before": "Similarity B determinant ORIGINAL second cross product := mul(0, 11)",
      "after": "Similarity B determinant ORIGINAL second cross product = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本題原始角色作精確有理運算；SAFE只約束原輸入，衍生整數不截斷，不作浮點近似。",
        "en": "Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Similarity B determinant ORIGINAL second cross product，不以小數近似替代。",
          "en": "Recompute Similarity B determinant ORIGINAL second cross product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "11"
        ],
        "output": "0",
        "label": "Similarity B determinant ORIGINAL second cross product"
      }
    },
    {
      "id": "S033",
      "uses": [
        "G001",
        "S032"
      ],
      "before": "Similarity B determinant ORIGINAL cross difference := sub(-726, 0)",
      "after": "Similarity B determinant ORIGINAL cross difference = -726",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本題原始角色作精確有理運算；SAFE只約束原輸入，衍生整數不截斷，不作浮點近似。",
        "en": "Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Similarity B determinant ORIGINAL cross difference，不以小數近似替代。",
          "en": "Recompute Similarity B determinant ORIGINAL cross difference with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "-726",
          "0"
        ],
        "output": "-726",
        "label": "Similarity B determinant ORIGINAL cross difference"
      }
    },
    {
      "id": "S034",
      "uses": [
        "G001",
        "S033"
      ],
      "before": "source roles = [Similarity B, [[3, -4], [36, -4], [14, -26]]]",
      "after": "derived = [-726, 是]",
      "relation": "implies",
      "rule_id": "Similarity_ACTUAL_proper_ordered_triangle_no_relabel",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先驗兩邊各為真nondegenerate triangle：兩條基點向量cross不等0。保原A1,A2,A3或B1,B2,B3次序，不能重排找另一個配對。",
        "en": "Require an ACTUAL nondegenerate triangle via the nonzero cross product of two basepoint vectors.Retain original A1,A2,A3 or B1,B2,B3 labels;do not search another correspondence."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向逐項代回原始資料與本步條件；保持索引、符號、嚴格／非嚴格邊界，不能用正確答案代替推導責任。這是候選教學與精確核對，非形式證明或發布資格。",
          "en": "Reverse-check EACH original datum and premise,retaining indices,signs and strict/non-strict boundaries.A correct final answer cannot replace the derivation obligations.This is candidate teaching/exact checking,not formal proof or publication clearance."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S035",
      "uses": [
        "G001",
        "S034"
      ],
      "before": "Similarity B side12 ORIGINAL coordinate0 difference := sub(36, 3)",
      "after": "Similarity B side12 ORIGINAL coordinate0 difference = 33",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本題原始角色作精確有理運算；SAFE只約束原輸入，衍生整數不截斷，不作浮點近似。",
        "en": "Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Similarity B side12 ORIGINAL coordinate0 difference，不以小數近似替代。",
          "en": "Recompute Similarity B side12 ORIGINAL coordinate0 difference with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "36",
          "3"
        ],
        "output": "33",
        "label": "Similarity B side12 ORIGINAL coordinate0 difference"
      }
    },
    {
      "id": "S036",
      "uses": [
        "G001",
        "S035"
      ],
      "before": "Similarity B side12 ORIGINAL coordinate1 difference := sub(-4, -4)",
      "after": "Similarity B side12 ORIGINAL coordinate1 difference = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本題原始角色作精確有理運算；SAFE只約束原輸入，衍生整數不截斷，不作浮點近似。",
        "en": "Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Similarity B side12 ORIGINAL coordinate1 difference，不以小數近似替代。",
          "en": "Recompute Similarity B side12 ORIGINAL coordinate1 difference with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "-4",
          "-4"
        ],
        "output": "0",
        "label": "Similarity B side12 ORIGINAL coordinate1 difference"
      }
    },
    {
      "id": "S037",
      "uses": [
        "G001",
        "S036"
      ],
      "before": "Similarity B side12 ORIGINAL coordinate0 square := mul(33, 33)",
      "after": "Similarity B side12 ORIGINAL coordinate0 square = 1089",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本題原始角色作精確有理運算；SAFE只約束原輸入，衍生整數不截斷，不作浮點近似。",
        "en": "Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Similarity B side12 ORIGINAL coordinate0 square，不以小數近似替代。",
          "en": "Recompute Similarity B side12 ORIGINAL coordinate0 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "33",
          "33"
        ],
        "output": "1089",
        "label": "Similarity B side12 ORIGINAL coordinate0 square"
      }
    },
    {
      "id": "S038",
      "uses": [
        "G001",
        "S037"
      ],
      "before": "Similarity B side12 ORIGINAL coordinate1 square := mul(0, 0)",
      "after": "Similarity B side12 ORIGINAL coordinate1 square = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本題原始角色作精確有理運算；SAFE只約束原輸入，衍生整數不截斷，不作浮點近似。",
        "en": "Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Similarity B side12 ORIGINAL coordinate1 square，不以小數近似替代。",
          "en": "Recompute Similarity B side12 ORIGINAL coordinate1 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "0"
        ],
        "output": "0",
        "label": "Similarity B side12 ORIGINAL coordinate1 square"
      }
    },
    {
      "id": "S039",
      "uses": [
        "G001",
        "S038"
      ],
      "before": "Similarity B side12 ORIGINAL squared length := sum(1089, 0)",
      "after": "Similarity B side12 ORIGINAL squared length = 1089",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本題原始角色作精確有理運算；SAFE只約束原輸入，衍生整數不截斷，不作浮點近似。",
        "en": "Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Similarity B side12 ORIGINAL squared length，不以小數近似替代。",
          "en": "Recompute Similarity B side12 ORIGINAL squared length with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1089",
          "0"
        ],
        "output": "1089",
        "label": "Similarity B side12 ORIGINAL squared length"
      }
    },
    {
      "id": "S040",
      "uses": [
        "G001",
        "S039"
      ],
      "before": "Similarity B side13 ORIGINAL coordinate0 difference := sub(14, 3)",
      "after": "Similarity B side13 ORIGINAL coordinate0 difference = 11",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本題原始角色作精確有理運算；SAFE只約束原輸入，衍生整數不截斷，不作浮點近似。",
        "en": "Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Similarity B side13 ORIGINAL coordinate0 difference，不以小數近似替代。",
          "en": "Recompute Similarity B side13 ORIGINAL coordinate0 difference with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "14",
          "3"
        ],
        "output": "11",
        "label": "Similarity B side13 ORIGINAL coordinate0 difference"
      }
    },
    {
      "id": "S041",
      "uses": [
        "G001",
        "S040"
      ],
      "before": "Similarity B side13 ORIGINAL coordinate1 difference := sub(-26, -4)",
      "after": "Similarity B side13 ORIGINAL coordinate1 difference = -22",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本題原始角色作精確有理運算；SAFE只約束原輸入，衍生整數不截斷，不作浮點近似。",
        "en": "Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Similarity B side13 ORIGINAL coordinate1 difference，不以小數近似替代。",
          "en": "Recompute Similarity B side13 ORIGINAL coordinate1 difference with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "-26",
          "-4"
        ],
        "output": "-22",
        "label": "Similarity B side13 ORIGINAL coordinate1 difference"
      }
    },
    {
      "id": "S042",
      "uses": [
        "G001",
        "S041"
      ],
      "before": "Similarity B side13 ORIGINAL coordinate0 square := mul(11, 11)",
      "after": "Similarity B side13 ORIGINAL coordinate0 square = 121",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本題原始角色作精確有理運算；SAFE只約束原輸入，衍生整數不截斷，不作浮點近似。",
        "en": "Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Similarity B side13 ORIGINAL coordinate0 square，不以小數近似替代。",
          "en": "Recompute Similarity B side13 ORIGINAL coordinate0 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "11",
          "11"
        ],
        "output": "121",
        "label": "Similarity B side13 ORIGINAL coordinate0 square"
      }
    },
    {
      "id": "S043",
      "uses": [
        "G001",
        "S042"
      ],
      "before": "Similarity B side13 ORIGINAL coordinate1 square := mul(-22, -22)",
      "after": "Similarity B side13 ORIGINAL coordinate1 square = 484",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本題原始角色作精確有理運算；SAFE只約束原輸入，衍生整數不截斷，不作浮點近似。",
        "en": "Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Similarity B side13 ORIGINAL coordinate1 square，不以小數近似替代。",
          "en": "Recompute Similarity B side13 ORIGINAL coordinate1 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-22",
          "-22"
        ],
        "output": "484",
        "label": "Similarity B side13 ORIGINAL coordinate1 square"
      }
    },
    {
      "id": "S044",
      "uses": [
        "G001",
        "S043"
      ],
      "before": "Similarity B side13 ORIGINAL squared length := sum(121, 484)",
      "after": "Similarity B side13 ORIGINAL squared length = 605",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本題原始角色作精確有理運算；SAFE只約束原輸入，衍生整數不截斷，不作浮點近似。",
        "en": "Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Similarity B side13 ORIGINAL squared length，不以小數近似替代。",
          "en": "Recompute Similarity B side13 ORIGINAL squared length with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "121",
          "484"
        ],
        "output": "605",
        "label": "Similarity B side13 ORIGINAL squared length"
      }
    },
    {
      "id": "S045",
      "uses": [
        "G001",
        "S044"
      ],
      "before": "Similarity B side23 ORIGINAL coordinate0 difference := sub(14, 36)",
      "after": "Similarity B side23 ORIGINAL coordinate0 difference = -22",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本題原始角色作精確有理運算；SAFE只約束原輸入，衍生整數不截斷，不作浮點近似。",
        "en": "Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Similarity B side23 ORIGINAL coordinate0 difference，不以小數近似替代。",
          "en": "Recompute Similarity B side23 ORIGINAL coordinate0 difference with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "14",
          "36"
        ],
        "output": "-22",
        "label": "Similarity B side23 ORIGINAL coordinate0 difference"
      }
    },
    {
      "id": "S046",
      "uses": [
        "G001",
        "S045"
      ],
      "before": "Similarity B side23 ORIGINAL coordinate1 difference := sub(-26, -4)",
      "after": "Similarity B side23 ORIGINAL coordinate1 difference = -22",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本題原始角色作精確有理運算；SAFE只約束原輸入，衍生整數不截斷，不作浮點近似。",
        "en": "Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Similarity B side23 ORIGINAL coordinate1 difference，不以小數近似替代。",
          "en": "Recompute Similarity B side23 ORIGINAL coordinate1 difference with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "-26",
          "-4"
        ],
        "output": "-22",
        "label": "Similarity B side23 ORIGINAL coordinate1 difference"
      }
    },
    {
      "id": "S047",
      "uses": [
        "G001",
        "S046"
      ],
      "before": "Similarity B side23 ORIGINAL coordinate0 square := mul(-22, -22)",
      "after": "Similarity B side23 ORIGINAL coordinate0 square = 484",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本題原始角色作精確有理運算；SAFE只約束原輸入，衍生整數不截斷，不作浮點近似。",
        "en": "Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Similarity B side23 ORIGINAL coordinate0 square，不以小數近似替代。",
          "en": "Recompute Similarity B side23 ORIGINAL coordinate0 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-22",
          "-22"
        ],
        "output": "484",
        "label": "Similarity B side23 ORIGINAL coordinate0 square"
      }
    },
    {
      "id": "S048",
      "uses": [
        "G001",
        "S047"
      ],
      "before": "Similarity B side23 ORIGINAL coordinate1 square := mul(-22, -22)",
      "after": "Similarity B side23 ORIGINAL coordinate1 square = 484",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本題原始角色作精確有理運算；SAFE只約束原輸入，衍生整數不截斷，不作浮點近似。",
        "en": "Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Similarity B side23 ORIGINAL coordinate1 square，不以小數近似替代。",
          "en": "Recompute Similarity B side23 ORIGINAL coordinate1 square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-22",
          "-22"
        ],
        "output": "484",
        "label": "Similarity B side23 ORIGINAL coordinate1 square"
      }
    },
    {
      "id": "S049",
      "uses": [
        "G001",
        "S048"
      ],
      "before": "Similarity B side23 ORIGINAL squared length := sum(484, 484)",
      "after": "Similarity B side23 ORIGINAL squared length = 968",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本題原始角色作精確有理運算；SAFE只約束原輸入，衍生整數不截斷，不作浮點近似。",
        "en": "Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Similarity B side23 ORIGINAL squared length，不以小數近似替代。",
          "en": "Recompute Similarity B side23 ORIGINAL squared length with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "484",
          "484"
        ],
        "output": "968",
        "label": "Similarity B side23 ORIGINAL squared length"
      }
    },
    {
      "id": "S050",
      "uses": [
        "G001",
        "S049"
      ],
      "before": "Similarity fixed side0 ORIGINAL Bsquare times q squared := mul(1089, 1)",
      "after": "Similarity fixed side0 ORIGINAL Bsquare times q squared = 1089",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本題原始角色作精確有理運算；SAFE只約束原輸入，衍生整數不截斷，不作浮點近似。",
        "en": "Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Similarity fixed side0 ORIGINAL Bsquare times q squared，不以小數近似替代。",
          "en": "Recompute Similarity fixed side0 ORIGINAL Bsquare times q squared with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1089",
          "1"
        ],
        "output": "1089",
        "label": "Similarity fixed side0 ORIGINAL Bsquare times q squared"
      }
    },
    {
      "id": "S051",
      "uses": [
        "G001",
        "S050"
      ],
      "before": "Similarity fixed side0 ORIGINAL Asquare times p squared := mul(9, 121)",
      "after": "Similarity fixed side0 ORIGINAL Asquare times p squared = 1089",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本題原始角色作精確有理運算；SAFE只約束原輸入，衍生整數不截斷，不作浮點近似。",
        "en": "Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Similarity fixed side0 ORIGINAL Asquare times p squared，不以小數近似替代。",
          "en": "Recompute Similarity fixed side0 ORIGINAL Asquare times p squared with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "9",
          "121"
        ],
        "output": "1089",
        "label": "Similarity fixed side0 ORIGINAL Asquare times p squared"
      }
    },
    {
      "id": "S052",
      "uses": [
        "G001",
        "S051"
      ],
      "before": "source roles = [0, 9, 1089, 11, 1]",
      "after": "derived = [1089, 1089, 是]",
      "relation": "implies",
      "rule_id": "Similarity_EACH_fixed_corresponding_squared_side_equality",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以給定forward scale從A到B，兩邊平方長為正，B²*q²=A²*p² iff B/A=p/q；每條12,13,23都要驗，不只第一邊。",
        "en": "Use the given FORWARD scale from A to B.Positive lengths give B²*q²=A²*p² IFF B/A=p/q.Check EVERY side12,13,23,not merely the first."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向逐項代回原始資料與本步條件；保持索引、符號、嚴格／非嚴格邊界，不能用正確答案代替推導責任。這是候選教學與精確核對，非形式證明或發布資格。",
          "en": "Reverse-check EACH original datum and premise,retaining indices,signs and strict/non-strict boundaries.A correct final answer cannot replace the derivation obligations.This is candidate teaching/exact checking,not formal proof or publication clearance."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S053",
      "uses": [
        "G001",
        "S052"
      ],
      "before": "Similarity fixed side1 ORIGINAL Bsquare times q squared := mul(605, 1)",
      "after": "Similarity fixed side1 ORIGINAL Bsquare times q squared = 605",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本題原始角色作精確有理運算；SAFE只約束原輸入，衍生整數不截斷，不作浮點近似。",
        "en": "Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Similarity fixed side1 ORIGINAL Bsquare times q squared，不以小數近似替代。",
          "en": "Recompute Similarity fixed side1 ORIGINAL Bsquare times q squared with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "605",
          "1"
        ],
        "output": "605",
        "label": "Similarity fixed side1 ORIGINAL Bsquare times q squared"
      }
    },
    {
      "id": "S054",
      "uses": [
        "G001",
        "S053"
      ],
      "before": "Similarity fixed side1 ORIGINAL Asquare times p squared := mul(5, 121)",
      "after": "Similarity fixed side1 ORIGINAL Asquare times p squared = 605",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本題原始角色作精確有理運算；SAFE只約束原輸入，衍生整數不截斷，不作浮點近似。",
        "en": "Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Similarity fixed side1 ORIGINAL Asquare times p squared，不以小數近似替代。",
          "en": "Recompute Similarity fixed side1 ORIGINAL Asquare times p squared with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "5",
          "121"
        ],
        "output": "605",
        "label": "Similarity fixed side1 ORIGINAL Asquare times p squared"
      }
    },
    {
      "id": "S055",
      "uses": [
        "G001",
        "S054"
      ],
      "before": "source roles = [1, 5, 605, 11, 1]",
      "after": "derived = [605, 605, 是]",
      "relation": "implies",
      "rule_id": "Similarity_EACH_fixed_corresponding_squared_side_equality",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以給定forward scale從A到B，兩邊平方長為正，B²*q²=A²*p² iff B/A=p/q；每條12,13,23都要驗，不只第一邊。",
        "en": "Use the given FORWARD scale from A to B.Positive lengths give B²*q²=A²*p² IFF B/A=p/q.Check EVERY side12,13,23,not merely the first."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向逐項代回原始資料與本步條件；保持索引、符號、嚴格／非嚴格邊界，不能用正確答案代替推導責任。這是候選教學與精確核對，非形式證明或發布資格。",
          "en": "Reverse-check EACH original datum and premise,retaining indices,signs and strict/non-strict boundaries.A correct final answer cannot replace the derivation obligations.This is candidate teaching/exact checking,not formal proof or publication clearance."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S056",
      "uses": [
        "G001",
        "S055"
      ],
      "before": "Similarity fixed side2 ORIGINAL Bsquare times q squared := mul(968, 1)",
      "after": "Similarity fixed side2 ORIGINAL Bsquare times q squared = 968",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本題原始角色作精確有理運算；SAFE只約束原輸入，衍生整數不截斷，不作浮點近似。",
        "en": "Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Similarity fixed side2 ORIGINAL Bsquare times q squared，不以小數近似替代。",
          "en": "Recompute Similarity fixed side2 ORIGINAL Bsquare times q squared with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "968",
          "1"
        ],
        "output": "968",
        "label": "Similarity fixed side2 ORIGINAL Bsquare times q squared"
      }
    },
    {
      "id": "S057",
      "uses": [
        "G001",
        "S056"
      ],
      "before": "Similarity fixed side2 ORIGINAL Asquare times p squared := mul(8, 121)",
      "after": "Similarity fixed side2 ORIGINAL Asquare times p squared = 968",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本題原始角色作精確有理運算；SAFE只約束原輸入，衍生整數不截斷，不作浮點近似。",
        "en": "Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Similarity fixed side2 ORIGINAL Asquare times p squared，不以小數近似替代。",
          "en": "Recompute Similarity fixed side2 ORIGINAL Asquare times p squared with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "8",
          "121"
        ],
        "output": "968",
        "label": "Similarity fixed side2 ORIGINAL Asquare times p squared"
      }
    },
    {
      "id": "S058",
      "uses": [
        "G001",
        "S057"
      ],
      "before": "source roles = [2, 8, 968, 11, 1]",
      "after": "derived = [968, 968, 是]",
      "relation": "implies",
      "rule_id": "Similarity_EACH_fixed_corresponding_squared_side_equality",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以給定forward scale從A到B，兩邊平方長為正，B²*q²=A²*p² iff B/A=p/q；每條12,13,23都要驗，不只第一邊。",
        "en": "Use the given FORWARD scale from A to B.Positive lengths give B²*q²=A²*p² IFF B/A=p/q.Check EVERY side12,13,23,not merely the first."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向逐項代回原始資料與本步條件；保持索引、符號、嚴格／非嚴格邊界，不能用正確答案代替推導責任。這是候選教學與精確核對，非形式證明或發布資格。",
          "en": "Reverse-check EACH original datum and premise,retaining indices,signs and strict/non-strict boundaries.A correct final answer cannot replace the derivation obligations.This is candidate teaching/exact checking,not formal proof or publication clearance."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S059",
      "uses": [
        "G001",
        "S058"
      ],
      "before": "source roles = [[[0, 0], [3, 0], [1, 2]], [[3, -4], [36, -4], [14, -26]], 11, 1]",
      "after": "derived = [[是, 是, 是], 是, 是]",
      "relation": "implies",
      "rule_id": "Similarity_SSS_ALL_three_FIXED_correspondence_and_given_scale",
      "conditions": [],
      "explanation": {
        "zh-Hant": "真三角形的三個對應邊同正比才給SSS相似；含鏡像也可相似。No只否定此對應及給定forward比，不能擴成任意比或排列都不相似。",
        "en": "For proper triangles,ALL three corresponding sides at the same positive scale give SSS similarity,including a reflected copy.No rejects THIS correspondence and forward scale,not every possible ratio or permutation."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向逐項代回原始資料與本步條件；保持索引、符號、嚴格／非嚴格邊界，不能用正確答案代替推導責任。這是候選教學與精確核對，非形式證明或發布資格。",
          "en": "Reverse-check EACH original datum and premise,retaining indices,signs and strict/non-strict boundaries.A correct final answer cannot replace the derivation obligations.This is candidate teaching/exact checking,not formal proof or publication clearance."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "result": "是"
  },
  "conclusion": {
    "canonical_value": "是",
    "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-r06-similarity-foundations",
      "batch_path": "public/data/questions/batches/batch-20261006-advance-20261007-r06-similarity-foundations/questions.jsonl",
      "batch_sha256": "85dde18b8aa8a90c777016e6a02bf75e3032d389b8b3a625c6c5ec208949a4f6",
      "original_line": 11,
      "original_line_sha256": "7cc0292956c718eed2a6908cc7a0243fecbe76e88e1a66caf08620a6a341ae46",
      "question_compact_sha256": "aa0d04f7f11cea00a3d91d02f2b6f2c11db37dbd4cb6d2a07eebc9a50670bce5",
      "english_row_compact_sha256": "96d5f3fe9a9d9c862097626e5ecb10f3d5250a3fc37c09fb6f7eac128bb44dda",
      "english_manifest_path": "public/data/i18n/editions/batch-20261006-advance-20261007-r06-similarity-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261006-advance-20261007-r06-similarity-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-similarity": {
        "sha256": "d60b59d396063a61d7b04c2039c88873846f8e3486fe0fa326adf55123c006f4",
        "english_sha256": "a5cdadef7a5b49238b9ced0b42ab794c51fdffe9bd83f46239b1c56d962af72e",
        "original_path": "public/data/mko/similarity.json",
        "english_path": "public/data/i18n/editions/mko-similarity.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "e340023047850507ee3fe2ae1466d606cfcbd26849aa3bb168d53248f38097e5",
    "original_candidate_sha256": "4fea265ae25d40b5c413f692cd56b909e8972f5aee46fd0b87dcaf3ca2ec1506",
    "original_candidate_line_sha256": "4fea265ae25d40b5c413f692cd56b909e8972f5aee46fd0b87dcaf3ca2ec1506"
  },
  "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-20261026-010-p57",
    "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": "8bbda4edbed7078634ae88342998c79bcfb95de96187fac119c3b08e0adb0bc8",
      "question_first_witness_sha256": "3edafa72d63d16a22414892b55806430e0a25b15b926d04f312550179607759f"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 59,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "04d3f7b1b53a2175e0fd8707acf3c0dbdc2b84c660bca89bf8f716e25f5ad050",
      "arithmetic_steps": 52
    },
    "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."
  }
}