{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0066",
  "pilot_number": "R0047-24191",
  "number": 24191,
  "family_id": "qf-mod44-eulercrit-residue-nonresidue-cases",
  "title": {
    "zh-Hant": "第 24191 題：Euler二次剩餘判準",
    "en": "Question 24191: Residues Give Plus One and Nonresidues Minus One under the Qualified Criterion"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0066",
    "batch_id": "batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-euler-criterion-quadratic-residue"
    ],
    "title_zh": "Euler二次剩餘判準",
    "stem_zh": "odd prime、nonzero class、complete square table與independent repeated-squaring。操作=eulercrit-residue-nonresidue-cases；資料={\"prime\":59,\"a\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(59, 2, quadratic_nonresidue, [], -1, -1, 是, nonresidue_maps_to_minus_one)",
      "explanation_zh": "Wave6第七層核對=(59, 2, quadratic_nonresidue, [], -1, -1, 是, nonresidue_maps_to_minus_one)；依完整明示資料獨立重算；Legendre值由square table先獨立取得，p-1代表-1；finite table只核指定instance而非一般定理證明；p=2、composite與zero class不在domain；不得用Euler criterion反定義Legendre。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 2,
      "skills": [
        "Euler二次剩餘判準"
      ]
    },
    "variant": {
      "family_id": "qf-mod44-eulercrit-residue-nonresidue-cases",
      "seed": 202611134066,
      "parameters": {
        "prime": 59,
        "a": 2,
        "domain": "Odd_prime_Euler_criterion",
        "operation": "eulercrit-residue-nonresidue-cases",
        "semantic_template": "qf-mod44-eulercrit-residue-nonresidue-cases/{\"a\":2,\"prime\":59}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_finite_or_symbolic_model",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-15",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0066",
    "batch_id": "batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "1ab00abb71a1219e7d788b552be95da97b5e52d80ca6a3c844030e8003538628",
      "family_id": "qf-mod44-eulercrit-residue-nonresidue-cases",
      "seed": 202611134066,
      "semantic_template": "qf-mod44-eulercrit-residue-nonresidue-cases/{\"a\":2,\"prime\":59}"
    },
    "fields": {
      "title": "Residues Give Plus One and Nonresidues Minus One under the Qualified Criterion",
      "stem": "Use odd prime p=59, original a=2, normalized nonzero class. Give (prime, normalized_class, classification, nonzero_roots, Legendre_symbol, signed_power, equal, certificate).",
      "answer": {
        "kind": "expression",
        "value": "(59, 2, quadratic_nonresidue, [], -1, -1, Yes, nonresidue_maps_to_minus_one)",
        "explanation": "Keep square-table classification, roots, Legendre value and signed half-power result separate. Modulo-p representative p−1 means−1, not integer equality of−1 and p−1. No Jacobi or composite-modulus converse is asserted. The exact result, retaining the requested order, is (59, 2, quadratic_nonresidue, [], -1, -1, Yes, nonresidue_maps_to_minus_one)."
      },
      "skills": [
        "Residues Give Plus One and Nonresidues Minus One under the Qualified Criterion"
      ],
      "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": "odd prime、nonzero class、complete square table與independent repeated-squaring。操作=eulercrit-residue-nonresidue-cases；資料={\"prime\":59,\"a\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
        "en": "Use odd prime p=59, original a=2, normalized nonzero class. Give (prime, normalized_class, classification, nonzero_roots, Legendre_symbol, signed_power, equal, certificate)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source roles = [eulercrit-residue-nonresidue-cases]",
      "after": "derived = [[prime, normalized_class, classification, nonzero_roots, Legendre_symbol, signed_power, equal, certificate]]",
      "relation": "implies",
      "rule_id": "source_ordered_fields_without_JS",
      "conditions": [],
      "explanation": {
        "zh-Hant": "本頁列完整來源欄序，不需JavaScript；gradient boundary依原mode選6／8／7欄，不補欄也不由答案猜mode。",
        "en": "List the source field order without JavaScript.Gradient boundary uses its original mode to choose6,8 or7 fields;no padding or answer-derived mode."
      },
      "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": "source roles = [59]",
      "after": "derived = [[[2, 1], [3, 2], [4, 3], [5, 4], [6, 5], [7, 3]], 是]",
      "relation": "implies",
      "rule_id": "Euler_modulus_primality_complete_trial",
      "conditions": [],
      "explanation": {
        "zh-Hant": "合數必有不大於平方根的因數，故完整試除驗模數資格；這不由平方性或Euler等式猜測。",
        "en": "A composite has a factor at most its square root;complete trial division qualifies the modulus,not a guessed square/Euler equality."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "Euler ORIGINAL signed quotient times modulus := mul(0, 59)",
      "after": "Euler ORIGINAL signed quotient times modulus = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euler ORIGINAL signed quotient times modulus，不以小數近似替代。",
          "en": "Recompute Euler ORIGINAL signed quotient times modulus with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "59"
        ],
        "output": "0",
        "label": "Euler ORIGINAL signed quotient times modulus"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "Euler ORIGINAL signed a minus modulus multiple := sub(2, 0)",
      "after": "Euler ORIGINAL signed a minus modulus multiple = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euler ORIGINAL signed a minus modulus multiple，不以小數近似替代。",
          "en": "Recompute Euler ORIGINAL signed a minus modulus multiple with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "2",
          "0"
        ],
        "output": "2",
        "label": "Euler ORIGINAL signed a minus modulus multiple"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "source roles = [2, 59, 0]",
      "after": "derived = [2, 是]",
      "relation": "implies",
      "rule_id": "Euler_Euclidean_class_before_hypothesis",
      "conditions": [],
      "explanation": {
        "zh-Hant": "a=qp+r、0<=r<p保負a同一類；原a在指定欄位保留。",
        "en": "Use a=qp+r with0<=r<p,including negative a;retain original a where the interface requests it."
      },
      "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": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "source roles = [59, 2, 2]",
      "after": "derived = [是, 是]",
      "relation": "implies",
      "rule_id": "Euler_odd_prime_nonzero_qualified",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先獨立驗odd prime與非零類；平方表先定義Legendre，再比較冪。",
        "en": "Qualify odd primality and the nonzero class first.Define Legendre by the square table before comparing a power."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "Euler square root candidate1 := mul(1, 1)",
      "after": "Euler square root candidate1 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euler square root candidate1，不以小數近似替代。",
          "en": "Recompute Euler square root candidate1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "1"
        ],
        "output": "1",
        "label": "Euler square root candidate1"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "Euler square root candidate2 := mul(2, 2)",
      "after": "Euler square root candidate2 = 4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euler square root candidate2，不以小數近似替代。",
          "en": "Recompute Euler square root candidate2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "2"
        ],
        "output": "4",
        "label": "Euler square root candidate2"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "Euler square root candidate3 := mul(3, 3)",
      "after": "Euler square root candidate3 = 9",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euler square root candidate3，不以小數近似替代。",
          "en": "Recompute Euler square root candidate3 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "3",
          "3"
        ],
        "output": "9",
        "label": "Euler square root candidate3"
      }
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "Euler square root candidate4 := mul(4, 4)",
      "after": "Euler square root candidate4 = 16",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euler square root candidate4，不以小數近似替代。",
          "en": "Recompute Euler square root candidate4 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "4",
          "4"
        ],
        "output": "16",
        "label": "Euler square root candidate4"
      }
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "Euler square root candidate5 := mul(5, 5)",
      "after": "Euler square root candidate5 = 25",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euler square root candidate5，不以小數近似替代。",
          "en": "Recompute Euler square root candidate5 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "5",
          "5"
        ],
        "output": "25",
        "label": "Euler square root candidate5"
      }
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "Euler square root candidate6 := mul(6, 6)",
      "after": "Euler square root candidate6 = 36",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euler square root candidate6，不以小數近似替代。",
          "en": "Recompute Euler square root candidate6 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "6",
          "6"
        ],
        "output": "36",
        "label": "Euler square root candidate6"
      }
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "Euler square root candidate7 := mul(7, 7)",
      "after": "Euler square root candidate7 = 49",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euler square root candidate7，不以小數近似替代。",
          "en": "Recompute Euler square root candidate7 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "7",
          "7"
        ],
        "output": "49",
        "label": "Euler square root candidate7"
      }
    },
    {
      "id": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "Euler square root candidate8 := mul(8, 8)",
      "after": "Euler square root candidate8 = 64",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euler square root candidate8，不以小數近似替代。",
          "en": "Recompute Euler square root candidate8 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "8",
          "8"
        ],
        "output": "64",
        "label": "Euler square root candidate8"
      }
    },
    {
      "id": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "Euler square root candidate9 := mul(9, 9)",
      "after": "Euler square root candidate9 = 81",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euler square root candidate9，不以小數近似替代。",
          "en": "Recompute Euler square root candidate9 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "9",
          "9"
        ],
        "output": "81",
        "label": "Euler square root candidate9"
      }
    },
    {
      "id": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "Euler square root candidate10 := mul(10, 10)",
      "after": "Euler square root candidate10 = 100",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euler square root candidate10，不以小數近似替代。",
          "en": "Recompute Euler square root candidate10 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "10",
          "10"
        ],
        "output": "100",
        "label": "Euler square root candidate10"
      }
    },
    {
      "id": "S017",
      "uses": [
        "G001",
        "S016"
      ],
      "before": "Euler square root candidate11 := mul(11, 11)",
      "after": "Euler square root candidate11 = 121",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euler square root candidate11，不以小數近似替代。",
          "en": "Recompute Euler square root candidate11 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "11",
          "11"
        ],
        "output": "121",
        "label": "Euler square root candidate11"
      }
    },
    {
      "id": "S018",
      "uses": [
        "G001",
        "S017"
      ],
      "before": "Euler square root candidate12 := mul(12, 12)",
      "after": "Euler square root candidate12 = 144",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euler square root candidate12，不以小數近似替代。",
          "en": "Recompute Euler square root candidate12 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "12",
          "12"
        ],
        "output": "144",
        "label": "Euler square root candidate12"
      }
    },
    {
      "id": "S019",
      "uses": [
        "G001",
        "S018"
      ],
      "before": "Euler square root candidate13 := mul(13, 13)",
      "after": "Euler square root candidate13 = 169",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euler square root candidate13，不以小數近似替代。",
          "en": "Recompute Euler square root candidate13 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "13",
          "13"
        ],
        "output": "169",
        "label": "Euler square root candidate13"
      }
    },
    {
      "id": "S020",
      "uses": [
        "G001",
        "S019"
      ],
      "before": "Euler square root candidate14 := mul(14, 14)",
      "after": "Euler square root candidate14 = 196",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euler square root candidate14，不以小數近似替代。",
          "en": "Recompute Euler square root candidate14 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "14",
          "14"
        ],
        "output": "196",
        "label": "Euler square root candidate14"
      }
    },
    {
      "id": "S021",
      "uses": [
        "G001",
        "S020"
      ],
      "before": "Euler square root candidate15 := mul(15, 15)",
      "after": "Euler square root candidate15 = 225",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euler square root candidate15，不以小數近似替代。",
          "en": "Recompute Euler square root candidate15 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "15",
          "15"
        ],
        "output": "225",
        "label": "Euler square root candidate15"
      }
    },
    {
      "id": "S022",
      "uses": [
        "G001",
        "S021"
      ],
      "before": "Euler square root candidate16 := mul(16, 16)",
      "after": "Euler square root candidate16 = 256",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euler square root candidate16，不以小數近似替代。",
          "en": "Recompute Euler square root candidate16 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "16",
          "16"
        ],
        "output": "256",
        "label": "Euler square root candidate16"
      }
    },
    {
      "id": "S023",
      "uses": [
        "G001",
        "S022"
      ],
      "before": "Euler square root candidate17 := mul(17, 17)",
      "after": "Euler square root candidate17 = 289",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euler square root candidate17，不以小數近似替代。",
          "en": "Recompute Euler square root candidate17 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "17",
          "17"
        ],
        "output": "289",
        "label": "Euler square root candidate17"
      }
    },
    {
      "id": "S024",
      "uses": [
        "G001",
        "S023"
      ],
      "before": "Euler square root candidate18 := mul(18, 18)",
      "after": "Euler square root candidate18 = 324",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euler square root candidate18，不以小數近似替代。",
          "en": "Recompute Euler square root candidate18 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "18",
          "18"
        ],
        "output": "324",
        "label": "Euler square root candidate18"
      }
    },
    {
      "id": "S025",
      "uses": [
        "G001",
        "S024"
      ],
      "before": "Euler square root candidate19 := mul(19, 19)",
      "after": "Euler square root candidate19 = 361",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euler square root candidate19，不以小數近似替代。",
          "en": "Recompute Euler square root candidate19 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "19",
          "19"
        ],
        "output": "361",
        "label": "Euler square root candidate19"
      }
    },
    {
      "id": "S026",
      "uses": [
        "G001",
        "S025"
      ],
      "before": "Euler square root candidate20 := mul(20, 20)",
      "after": "Euler square root candidate20 = 400",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euler square root candidate20，不以小數近似替代。",
          "en": "Recompute Euler square root candidate20 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "20",
          "20"
        ],
        "output": "400",
        "label": "Euler square root candidate20"
      }
    },
    {
      "id": "S027",
      "uses": [
        "G001",
        "S026"
      ],
      "before": "Euler square root candidate21 := mul(21, 21)",
      "after": "Euler square root candidate21 = 441",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euler square root candidate21，不以小數近似替代。",
          "en": "Recompute Euler square root candidate21 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "21",
          "21"
        ],
        "output": "441",
        "label": "Euler square root candidate21"
      }
    },
    {
      "id": "S028",
      "uses": [
        "G001",
        "S027"
      ],
      "before": "Euler square root candidate22 := mul(22, 22)",
      "after": "Euler square root candidate22 = 484",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euler square root candidate22，不以小數近似替代。",
          "en": "Recompute Euler square root candidate22 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "22",
          "22"
        ],
        "output": "484",
        "label": "Euler square root candidate22"
      }
    },
    {
      "id": "S029",
      "uses": [
        "G001",
        "S028"
      ],
      "before": "Euler square root candidate23 := mul(23, 23)",
      "after": "Euler square root candidate23 = 529",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euler square root candidate23，不以小數近似替代。",
          "en": "Recompute Euler square root candidate23 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "23",
          "23"
        ],
        "output": "529",
        "label": "Euler square root candidate23"
      }
    },
    {
      "id": "S030",
      "uses": [
        "G001",
        "S029"
      ],
      "before": "Euler square root candidate24 := mul(24, 24)",
      "after": "Euler square root candidate24 = 576",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euler square root candidate24，不以小數近似替代。",
          "en": "Recompute Euler square root candidate24 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "24",
          "24"
        ],
        "output": "576",
        "label": "Euler square root candidate24"
      }
    },
    {
      "id": "S031",
      "uses": [
        "G001",
        "S030"
      ],
      "before": "Euler square root candidate25 := mul(25, 25)",
      "after": "Euler square root candidate25 = 625",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euler square root candidate25，不以小數近似替代。",
          "en": "Recompute Euler square root candidate25 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "25"
        ],
        "output": "625",
        "label": "Euler square root candidate25"
      }
    },
    {
      "id": "S032",
      "uses": [
        "G001",
        "S031"
      ],
      "before": "Euler square root candidate26 := mul(26, 26)",
      "after": "Euler square root candidate26 = 676",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euler square root candidate26，不以小數近似替代。",
          "en": "Recompute Euler square root candidate26 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "26",
          "26"
        ],
        "output": "676",
        "label": "Euler square root candidate26"
      }
    },
    {
      "id": "S033",
      "uses": [
        "G001",
        "S032"
      ],
      "before": "Euler square root candidate27 := mul(27, 27)",
      "after": "Euler square root candidate27 = 729",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euler square root candidate27，不以小數近似替代。",
          "en": "Recompute Euler square root candidate27 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "27",
          "27"
        ],
        "output": "729",
        "label": "Euler square root candidate27"
      }
    },
    {
      "id": "S034",
      "uses": [
        "G001",
        "S033"
      ],
      "before": "Euler square root candidate28 := mul(28, 28)",
      "after": "Euler square root candidate28 = 784",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euler square root candidate28，不以小數近似替代。",
          "en": "Recompute Euler square root candidate28 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "28",
          "28"
        ],
        "output": "784",
        "label": "Euler square root candidate28"
      }
    },
    {
      "id": "S035",
      "uses": [
        "G001",
        "S034"
      ],
      "before": "Euler square root candidate29 := mul(29, 29)",
      "after": "Euler square root candidate29 = 841",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euler square root candidate29，不以小數近似替代。",
          "en": "Recompute Euler square root candidate29 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "29",
          "29"
        ],
        "output": "841",
        "label": "Euler square root candidate29"
      }
    },
    {
      "id": "S036",
      "uses": [
        "G001",
        "S035"
      ],
      "before": "Euler square root candidate30 := mul(30, 30)",
      "after": "Euler square root candidate30 = 900",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euler square root candidate30，不以小數近似替代。",
          "en": "Recompute Euler square root candidate30 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "30",
          "30"
        ],
        "output": "900",
        "label": "Euler square root candidate30"
      }
    },
    {
      "id": "S037",
      "uses": [
        "G001",
        "S036"
      ],
      "before": "Euler square root candidate31 := mul(31, 31)",
      "after": "Euler square root candidate31 = 961",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euler square root candidate31，不以小數近似替代。",
          "en": "Recompute Euler square root candidate31 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "31",
          "31"
        ],
        "output": "961",
        "label": "Euler square root candidate31"
      }
    },
    {
      "id": "S038",
      "uses": [
        "G001",
        "S037"
      ],
      "before": "Euler square root candidate32 := mul(32, 32)",
      "after": "Euler square root candidate32 = 1024",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euler square root candidate32，不以小數近似替代。",
          "en": "Recompute Euler square root candidate32 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "32",
          "32"
        ],
        "output": "1024",
        "label": "Euler square root candidate32"
      }
    },
    {
      "id": "S039",
      "uses": [
        "G001",
        "S038"
      ],
      "before": "Euler square root candidate33 := mul(33, 33)",
      "after": "Euler square root candidate33 = 1089",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euler square root candidate33，不以小數近似替代。",
          "en": "Recompute Euler square root candidate33 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "33",
          "33"
        ],
        "output": "1089",
        "label": "Euler square root candidate33"
      }
    },
    {
      "id": "S040",
      "uses": [
        "G001",
        "S039"
      ],
      "before": "Euler square root candidate34 := mul(34, 34)",
      "after": "Euler square root candidate34 = 1156",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euler square root candidate34，不以小數近似替代。",
          "en": "Recompute Euler square root candidate34 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "34",
          "34"
        ],
        "output": "1156",
        "label": "Euler square root candidate34"
      }
    },
    {
      "id": "S041",
      "uses": [
        "G001",
        "S040"
      ],
      "before": "Euler square root candidate35 := mul(35, 35)",
      "after": "Euler square root candidate35 = 1225",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euler square root candidate35，不以小數近似替代。",
          "en": "Recompute Euler square root candidate35 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "35",
          "35"
        ],
        "output": "1225",
        "label": "Euler square root candidate35"
      }
    },
    {
      "id": "S042",
      "uses": [
        "G001",
        "S041"
      ],
      "before": "Euler square root candidate36 := mul(36, 36)",
      "after": "Euler square root candidate36 = 1296",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euler square root candidate36，不以小數近似替代。",
          "en": "Recompute Euler square root candidate36 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "36",
          "36"
        ],
        "output": "1296",
        "label": "Euler square root candidate36"
      }
    },
    {
      "id": "S043",
      "uses": [
        "G001",
        "S042"
      ],
      "before": "Euler square root candidate37 := mul(37, 37)",
      "after": "Euler square root candidate37 = 1369",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euler square root candidate37，不以小數近似替代。",
          "en": "Recompute Euler square root candidate37 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "37",
          "37"
        ],
        "output": "1369",
        "label": "Euler square root candidate37"
      }
    },
    {
      "id": "S044",
      "uses": [
        "G001",
        "S043"
      ],
      "before": "Euler square root candidate38 := mul(38, 38)",
      "after": "Euler square root candidate38 = 1444",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euler square root candidate38，不以小數近似替代。",
          "en": "Recompute Euler square root candidate38 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "38",
          "38"
        ],
        "output": "1444",
        "label": "Euler square root candidate38"
      }
    },
    {
      "id": "S045",
      "uses": [
        "G001",
        "S044"
      ],
      "before": "Euler square root candidate39 := mul(39, 39)",
      "after": "Euler square root candidate39 = 1521",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euler square root candidate39，不以小數近似替代。",
          "en": "Recompute Euler square root candidate39 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "39",
          "39"
        ],
        "output": "1521",
        "label": "Euler square root candidate39"
      }
    },
    {
      "id": "S046",
      "uses": [
        "G001",
        "S045"
      ],
      "before": "Euler square root candidate40 := mul(40, 40)",
      "after": "Euler square root candidate40 = 1600",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euler square root candidate40，不以小數近似替代。",
          "en": "Recompute Euler square root candidate40 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "40",
          "40"
        ],
        "output": "1600",
        "label": "Euler square root candidate40"
      }
    },
    {
      "id": "S047",
      "uses": [
        "G001",
        "S046"
      ],
      "before": "Euler square root candidate41 := mul(41, 41)",
      "after": "Euler square root candidate41 = 1681",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euler square root candidate41，不以小數近似替代。",
          "en": "Recompute Euler square root candidate41 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "41",
          "41"
        ],
        "output": "1681",
        "label": "Euler square root candidate41"
      }
    },
    {
      "id": "S048",
      "uses": [
        "G001",
        "S047"
      ],
      "before": "Euler square root candidate42 := mul(42, 42)",
      "after": "Euler square root candidate42 = 1764",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euler square root candidate42，不以小數近似替代。",
          "en": "Recompute Euler square root candidate42 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "42",
          "42"
        ],
        "output": "1764",
        "label": "Euler square root candidate42"
      }
    },
    {
      "id": "S049",
      "uses": [
        "G001",
        "S048"
      ],
      "before": "Euler square root candidate43 := mul(43, 43)",
      "after": "Euler square root candidate43 = 1849",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euler square root candidate43，不以小數近似替代。",
          "en": "Recompute Euler square root candidate43 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "43",
          "43"
        ],
        "output": "1849",
        "label": "Euler square root candidate43"
      }
    },
    {
      "id": "S050",
      "uses": [
        "G001",
        "S049"
      ],
      "before": "Euler square root candidate44 := mul(44, 44)",
      "after": "Euler square root candidate44 = 1936",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euler square root candidate44，不以小數近似替代。",
          "en": "Recompute Euler square root candidate44 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "44",
          "44"
        ],
        "output": "1936",
        "label": "Euler square root candidate44"
      }
    },
    {
      "id": "S051",
      "uses": [
        "G001",
        "S050"
      ],
      "before": "Euler square root candidate45 := mul(45, 45)",
      "after": "Euler square root candidate45 = 2025",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euler square root candidate45，不以小數近似替代。",
          "en": "Recompute Euler square root candidate45 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "45",
          "45"
        ],
        "output": "2025",
        "label": "Euler square root candidate45"
      }
    },
    {
      "id": "S052",
      "uses": [
        "G001",
        "S051"
      ],
      "before": "Euler square root candidate46 := mul(46, 46)",
      "after": "Euler square root candidate46 = 2116",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euler square root candidate46，不以小數近似替代。",
          "en": "Recompute Euler square root candidate46 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "46",
          "46"
        ],
        "output": "2116",
        "label": "Euler square root candidate46"
      }
    },
    {
      "id": "S053",
      "uses": [
        "G001",
        "S052"
      ],
      "before": "Euler square root candidate47 := mul(47, 47)",
      "after": "Euler square root candidate47 = 2209",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euler square root candidate47，不以小數近似替代。",
          "en": "Recompute Euler square root candidate47 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "47",
          "47"
        ],
        "output": "2209",
        "label": "Euler square root candidate47"
      }
    },
    {
      "id": "S054",
      "uses": [
        "G001",
        "S053"
      ],
      "before": "Euler square root candidate48 := mul(48, 48)",
      "after": "Euler square root candidate48 = 2304",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euler square root candidate48，不以小數近似替代。",
          "en": "Recompute Euler square root candidate48 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "48",
          "48"
        ],
        "output": "2304",
        "label": "Euler square root candidate48"
      }
    },
    {
      "id": "S055",
      "uses": [
        "G001",
        "S054"
      ],
      "before": "Euler square root candidate49 := mul(49, 49)",
      "after": "Euler square root candidate49 = 2401",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euler square root candidate49，不以小數近似替代。",
          "en": "Recompute Euler square root candidate49 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "49",
          "49"
        ],
        "output": "2401",
        "label": "Euler square root candidate49"
      }
    },
    {
      "id": "S056",
      "uses": [
        "G001",
        "S055"
      ],
      "before": "Euler square root candidate50 := mul(50, 50)",
      "after": "Euler square root candidate50 = 2500",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euler square root candidate50，不以小數近似替代。",
          "en": "Recompute Euler square root candidate50 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "50",
          "50"
        ],
        "output": "2500",
        "label": "Euler square root candidate50"
      }
    },
    {
      "id": "S057",
      "uses": [
        "G001",
        "S056"
      ],
      "before": "Euler square root candidate51 := mul(51, 51)",
      "after": "Euler square root candidate51 = 2601",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euler square root candidate51，不以小數近似替代。",
          "en": "Recompute Euler square root candidate51 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "51",
          "51"
        ],
        "output": "2601",
        "label": "Euler square root candidate51"
      }
    },
    {
      "id": "S058",
      "uses": [
        "G001",
        "S057"
      ],
      "before": "Euler square root candidate52 := mul(52, 52)",
      "after": "Euler square root candidate52 = 2704",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euler square root candidate52，不以小數近似替代。",
          "en": "Recompute Euler square root candidate52 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "52",
          "52"
        ],
        "output": "2704",
        "label": "Euler square root candidate52"
      }
    },
    {
      "id": "S059",
      "uses": [
        "G001",
        "S058"
      ],
      "before": "Euler square root candidate53 := mul(53, 53)",
      "after": "Euler square root candidate53 = 2809",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euler square root candidate53，不以小數近似替代。",
          "en": "Recompute Euler square root candidate53 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "53",
          "53"
        ],
        "output": "2809",
        "label": "Euler square root candidate53"
      }
    },
    {
      "id": "S060",
      "uses": [
        "G001",
        "S059"
      ],
      "before": "Euler square root candidate54 := mul(54, 54)",
      "after": "Euler square root candidate54 = 2916",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euler square root candidate54，不以小數近似替代。",
          "en": "Recompute Euler square root candidate54 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "54",
          "54"
        ],
        "output": "2916",
        "label": "Euler square root candidate54"
      }
    },
    {
      "id": "S061",
      "uses": [
        "G001",
        "S060"
      ],
      "before": "Euler square root candidate55 := mul(55, 55)",
      "after": "Euler square root candidate55 = 3025",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euler square root candidate55，不以小數近似替代。",
          "en": "Recompute Euler square root candidate55 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "55",
          "55"
        ],
        "output": "3025",
        "label": "Euler square root candidate55"
      }
    },
    {
      "id": "S062",
      "uses": [
        "G001",
        "S061"
      ],
      "before": "Euler square root candidate56 := mul(56, 56)",
      "after": "Euler square root candidate56 = 3136",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euler square root candidate56，不以小數近似替代。",
          "en": "Recompute Euler square root candidate56 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "56",
          "56"
        ],
        "output": "3136",
        "label": "Euler square root candidate56"
      }
    },
    {
      "id": "S063",
      "uses": [
        "G001",
        "S062"
      ],
      "before": "Euler square root candidate57 := mul(57, 57)",
      "after": "Euler square root candidate57 = 3249",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euler square root candidate57，不以小數近似替代。",
          "en": "Recompute Euler square root candidate57 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "57",
          "57"
        ],
        "output": "3249",
        "label": "Euler square root candidate57"
      }
    },
    {
      "id": "S064",
      "uses": [
        "G001",
        "S063"
      ],
      "before": "Euler square root candidate58 := mul(58, 58)",
      "after": "Euler square root candidate58 = 3364",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euler square root candidate58，不以小數近似替代。",
          "en": "Recompute Euler square root candidate58 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "58",
          "58"
        ],
        "output": "3364",
        "label": "Euler square root candidate58"
      }
    },
    {
      "id": "S065",
      "uses": [
        "G001",
        "S064"
      ],
      "before": "source roles = [59, 2]",
      "after": "derived = [[[1, 1], [2, 4], [3, 9], [4, 16], [5, 25], [6, 36], [7, 49], [8, 5], [9, 22], [10, 41], [11, 3], [12, 26], [13, 51], [14, 19], [15, 48], [16, 20], [17, 53], [18, 29], [19, 7], [20, 46], [21, 28], [22, 12], [23, 57], [24, 45], [25, 35], [26, 27], [27, 21], [28, 17], [29, 15], [30, 15], [31, 17], [32, 21], [33, 27], [34, 35], [35, 45], [36, 57], [37, 12], [38, 28], [39, 46], [40, 7], [41, 29], [42, 53], [43, 20], [44, 48], [45, 19], [46, 51], [47, 26], [48, 3], [49, 41], [50, 22], [51, 5], [52, 49], [53, 36], [54, 25], [55, 16], [56, 9], [57, 4], [58, 1]], [], -1]",
      "relation": "implies",
      "rule_id": "Euler_ALL_nonzero_square_table_defines_Legendre",
      "conditions": [],
      "explanation": {
        "zh-Hant": "全部1至p−1逐平方列舉，根存在為+1、全無為−1；此路徑沒有用Euler判準算Legendre。",
        "en": "Enumerate every nonzero square;roots give+1 and complete absence gives−1.This route does not use Euler criterion to compute Legendre."
      },
      "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": "S066",
      "uses": [
        "G001",
        "S065"
      ],
      "before": "Euler binary power accumulate0 := mul(1, 2)",
      "after": "Euler binary power accumulate0 = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euler binary power accumulate0，不以小數近似替代。",
          "en": "Recompute Euler binary power accumulate0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "2"
        ],
        "output": "2",
        "label": "Euler binary power accumulate0"
      }
    },
    {
      "id": "S067",
      "uses": [
        "G001",
        "S066"
      ],
      "before": "Euler binary power square0 := mul(2, 2)",
      "after": "Euler binary power square0 = 4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euler binary power square0，不以小數近似替代。",
          "en": "Recompute Euler binary power square0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "2"
        ],
        "output": "4",
        "label": "Euler binary power square0"
      }
    },
    {
      "id": "S068",
      "uses": [
        "G001",
        "S067"
      ],
      "before": "source roles = [0, [1, 2, 29], 59, 2, 29]",
      "after": "derived = [2, 4, 2, 4, 14]",
      "relation": "implies",
      "rule_id": "Euler_repeated_squaring_original_exponent_invariant",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每步保acc*base^remaining同餘原a^((p−1)/2)；奇數先乘，平方底數、整除指數，直到0。與全部square表是不同計算。",
        "en": "Maintain acc*base^remaining congruent to the original half-power:accumulate an odd bit,square the base,and halve the exponent to0.This is separate from the square-table calculation."
      },
      "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": "S069",
      "uses": [
        "G001",
        "S068"
      ],
      "before": "Euler binary power square1 := mul(4, 4)",
      "after": "Euler binary power square1 = 16",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euler binary power square1，不以小數近似替代。",
          "en": "Recompute Euler binary power square1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "4",
          "4"
        ],
        "output": "16",
        "label": "Euler binary power square1"
      }
    },
    {
      "id": "S070",
      "uses": [
        "G001",
        "S069"
      ],
      "before": "source roles = [1, [2, 4, 14], 59, 2, 29]",
      "after": "derived = [null, 16, 2, 16, 7]",
      "relation": "implies",
      "rule_id": "Euler_repeated_squaring_original_exponent_invariant",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每步保acc*base^remaining同餘原a^((p−1)/2)；奇數先乘，平方底數、整除指數，直到0。與全部square表是不同計算。",
        "en": "Maintain acc*base^remaining congruent to the original half-power:accumulate an odd bit,square the base,and halve the exponent to0.This is separate from the square-table calculation."
      },
      "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": "S071",
      "uses": [
        "G001",
        "S070"
      ],
      "before": "Euler binary power accumulate2 := mul(2, 16)",
      "after": "Euler binary power accumulate2 = 32",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euler binary power accumulate2，不以小數近似替代。",
          "en": "Recompute Euler binary power accumulate2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "16"
        ],
        "output": "32",
        "label": "Euler binary power accumulate2"
      }
    },
    {
      "id": "S072",
      "uses": [
        "G001",
        "S071"
      ],
      "before": "Euler binary power square2 := mul(16, 16)",
      "after": "Euler binary power square2 = 256",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euler binary power square2，不以小數近似替代。",
          "en": "Recompute Euler binary power square2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "16",
          "16"
        ],
        "output": "256",
        "label": "Euler binary power square2"
      }
    },
    {
      "id": "S073",
      "uses": [
        "G001",
        "S072"
      ],
      "before": "source roles = [2, [2, 16, 7], 59, 2, 29]",
      "after": "derived = [32, 256, 32, 20, 3]",
      "relation": "implies",
      "rule_id": "Euler_repeated_squaring_original_exponent_invariant",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每步保acc*base^remaining同餘原a^((p−1)/2)；奇數先乘，平方底數、整除指數，直到0。與全部square表是不同計算。",
        "en": "Maintain acc*base^remaining congruent to the original half-power:accumulate an odd bit,square the base,and halve the exponent to0.This is separate from the square-table calculation."
      },
      "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": "S074",
      "uses": [
        "G001",
        "S073"
      ],
      "before": "Euler binary power accumulate3 := mul(32, 20)",
      "after": "Euler binary power accumulate3 = 640",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euler binary power accumulate3，不以小數近似替代。",
          "en": "Recompute Euler binary power accumulate3 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "32",
          "20"
        ],
        "output": "640",
        "label": "Euler binary power accumulate3"
      }
    },
    {
      "id": "S075",
      "uses": [
        "G001",
        "S074"
      ],
      "before": "Euler binary power square3 := mul(20, 20)",
      "after": "Euler binary power square3 = 400",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euler binary power square3，不以小數近似替代。",
          "en": "Recompute Euler binary power square3 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "20",
          "20"
        ],
        "output": "400",
        "label": "Euler binary power square3"
      }
    },
    {
      "id": "S076",
      "uses": [
        "G001",
        "S075"
      ],
      "before": "source roles = [3, [32, 20, 3], 59, 2, 29]",
      "after": "derived = [640, 400, 50, 46, 1]",
      "relation": "implies",
      "rule_id": "Euler_repeated_squaring_original_exponent_invariant",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每步保acc*base^remaining同餘原a^((p−1)/2)；奇數先乘，平方底數、整除指數，直到0。與全部square表是不同計算。",
        "en": "Maintain acc*base^remaining congruent to the original half-power:accumulate an odd bit,square the base,and halve the exponent to0.This is separate from the square-table calculation."
      },
      "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": "S077",
      "uses": [
        "G001",
        "S076"
      ],
      "before": "Euler binary power accumulate4 := mul(50, 46)",
      "after": "Euler binary power accumulate4 = 2300",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euler binary power accumulate4，不以小數近似替代。",
          "en": "Recompute Euler binary power accumulate4 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "50",
          "46"
        ],
        "output": "2300",
        "label": "Euler binary power accumulate4"
      }
    },
    {
      "id": "S078",
      "uses": [
        "G001",
        "S077"
      ],
      "before": "Euler binary power square4 := mul(46, 46)",
      "after": "Euler binary power square4 = 2116",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euler binary power square4，不以小數近似替代。",
          "en": "Recompute Euler binary power square4 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "46",
          "46"
        ],
        "output": "2116",
        "label": "Euler binary power square4"
      }
    },
    {
      "id": "S079",
      "uses": [
        "G001",
        "S078"
      ],
      "before": "source roles = [4, [50, 46, 1], 59, 2, 29]",
      "after": "derived = [2300, 2116, 58, 51, 0]",
      "relation": "implies",
      "rule_id": "Euler_repeated_squaring_original_exponent_invariant",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每步保acc*base^remaining同餘原a^((p−1)/2)；奇數先乘，平方底數、整除指數，直到0。與全部square表是不同計算。",
        "en": "Maintain acc*base^remaining congruent to the original half-power:accumulate an odd bit,square the base,and halve the exponent to0.This is separate from the square-table calculation."
      },
      "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": "S080",
      "uses": [
        "G001",
        "S079"
      ],
      "before": "source roles = [59, [], -1, 58]",
      "after": "derived = [-1, 是, one finite instance is not a general theorem proof]",
      "relation": "implies",
      "rule_id": "Euler_decode_p_minus1_and_instance_only_comparison",
      "conditions": [],
      "explanation": {
        "zh-Hant": "p−1是模p的−1代表，不是整數p−1等於−1。指定instance兩個獨立算法吻合，不把這一表當一般定理的證明。",
        "en": "p−1 represents−1 modulo p,not integer equality.Compare the two computations for this instance;one finite table is not a general theorem proof."
      },
      "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": [
      59,
      2,
      "quadratic_nonresidue",
      [],
      -1,
      -1,
      true,
      "nonresidue_maps_to_minus_one"
    ]
  },
  "conclusion": {
    "canonical_value": "(59, 2, quadratic_nonresidue, [], -1, -1, 是, nonresidue_maps_to_minus_one)",
    "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-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations",
      "batch_path": "public/data/questions/batches/batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations/questions.jsonl",
      "batch_sha256": "8896a9423d694eedac4e09e904a3a66db2c4856725c2c0bffd90fad44a2c65f5",
      "original_line": 66,
      "original_line_sha256": "d4a1ff4d5d806e727111be1ec27f3d822cb2228bb814452ab3b3e7ec254eccf7",
      "question_compact_sha256": "1ab00abb71a1219e7d788b552be95da97b5e52d80ca6a3c844030e8003538628",
      "english_row_compact_sha256": "d56470a85f72e42d10690a00c80e986569d52d1c15a011267a81c58845e7a92c",
      "english_manifest_path": "public/data/i18n/editions/batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-euler-criterion-quadratic-residue": {
        "sha256": "da968b0cd6fc238e595782d4d9588d4c8089b176b4d05fc3d1c01eccdb99ce68",
        "english_sha256": "6a1e1a855e1affb5b9eb585fa4ab2173f81281b4327bc8b562d2b308ee6bc99c",
        "original_path": "public/data/mko/euler-criterion-quadratic-residue.json",
        "english_path": "public/data/i18n/editions/mko-euler-criterion-quadratic-residue.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "b3aad195dad743751e4c8a4a6ce44f091c3824da1e52c21e9266d4ccd1dc1b68",
    "original_candidate_sha256": "2f21106318f525e9491984de5bfa40b6420f6a6db6676119aeaeef9c3f071964",
    "original_candidate_line_sha256": "2f21106318f525e9491984de5bfa40b6420f6a6db6676119aeaeef9c3f071964"
  },
  "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-20261026-009-p48",
    "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": "9b43b9823e9bd2478d6caa5b57ffed161c2c8956af86974e901d5662ad994386",
      "question_first_witness_sha256": "eb97596068e2d5648033bab2b8f6ca9b653bbe8cb77c0f8e1d16d0c40e46f50b"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 80,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "65efc30be8b4afeb49501cfb98b55d886e280a1a531d617fdaf4d6bad0714aeb",
      "arithmetic_steps": 69
    },
    "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."
  }
}