{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261014-advance-20261107-r06-euler-theorem-number-theory-0022",
  "pilot_number": "R0039-20022",
  "number": 20022,
  "family_id": "qf-mod38-eulerthm-congruence",
  "title": {
    "zh-Hant": "第 20022 題：Euler定理",
    "en": "Question 20022: Euler Congruence Requires a Unit Residue, Not a Prime Modulus"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261014-advance-20261107-r06-euler-theorem-number-theory-0022",
    "batch_id": "batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-euler-theorem-number-theory"
    ],
    "title_zh": "Euler定理",
    "stem_zh": "n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-congruence；資料={\"n\":26,\"a\":23}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(26, 23, 12, 1, 是)",
      "explanation_zh": "Wave6根層核對=(26, 23, 12, 1, 是)；依完整明示資料獨立重算；Euler定理要求gcd(a,n)=1並證a^phi(n)同餘1；負底數先正規化；nonunit不適用，單次congruence沒有primality converse；Fermat只是prime modulus特例。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 3,
      "skills": [
        "Euler定理"
      ]
    },
    "variant": {
      "family_id": "qf-mod38-eulerthm-congruence",
      "seed": 202611076022,
      "parameters": {
        "n": 26,
        "a": 23,
        "domain": "Exact_integer_Euler_theorem",
        "operation": "eulerthm-congruence",
        "semantic_template": "qf-mod38-eulerthm-congruence/{\"a\":23,\"n\":26}"
      }
    },
    "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、有限sample群作用、sup等於max、basis等於subbasis、signed Markov、pigeonhole converse、nonunit Euler、finite-prefix subsequence、Lean或Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-14",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261014-advance-20261107-r06-euler-theorem-number-theory-0022",
    "batch_id": "batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "c42477534741f16a7e77546618c67f071825ad0c35f1cdc365b2867f1c6b97b9",
      "family_id": "qf-mod38-eulerthm-congruence",
      "seed": 202611076022,
      "semantic_template": "qf-mod38-eulerthm-congruence/{\"a\":23,\"n\":26}"
    },
    "fields": {
      "title": "Euler Congruence Requires a Unit Residue, Not a Prime Modulus",
      "stem": "Use integer modulus n=26≥2 and original integer base a=23. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, normalized_base, totient, residue, congruent_to_one).",
      "answer": {
        "kind": "expression",
        "value": "(26, 23, 12, 1, Yes)",
        "explanation": "Require coprimality, compute phi(n) and a^phi(n) modulo n, then compare its residue with1. Composite moduli are allowed; Fermat is the prime-modulus special case. The exact result, retaining the requested order, is (26, 23, 12, 1, Yes)."
      },
      "skills": [
        "Euler Congruence Requires a Unit Residue, Not a Prime Modulus"
      ],
      "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": "n大於等於2、exact gcd、phi與安全modular exponentiation。操作=eulerthm-congruence；資料={\"n\":26,\"a\":23}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
        "en": "Use integer modulus n=26≥2 and original integer base a=23. Compute the positive normalized residue, gcd, Euler totient and modular power exactly by integer arithmetic. Euler's congruence requires gcd(a,n)=1 and is modulo n, not unrestricted integer equality; no primality conclusion follows from a passing congruence. Give (n, normalized_base, totient, residue, congruent_to_one)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source operation=eulerthm-congruence;original snapshot and mathematics retained",
      "after": "explicit ordered source output fields=[n, normalized_base, totient, residue, congruent_to_one]",
      "relation": "implies",
      "rule_id": "source_ordered_interface_explicit_without_javascript",
      "conditions": [],
      "explanation": {
        "zh-Hant": "原題快照保留歷史介面標記；本頁直接列來源欄位順序，不需JavaScript或外部表。欄位標籤不是額外數學證據，No保持原題的聲稱範圍。",
        "en": "Retain the historical interface marker in the original snapshot while listing ordered source fields here,without JavaScript or external lookup.Field labels are not additional mathematical evidence,and No retains the source claim scope."
      },
      "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": "Euler ORIGINAL signed base normalization exact quotient := div(23, 26)",
      "after": "Euler ORIGINAL signed base normalization exact quotient = 23/26",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留索引、正負號、分母與證明適用條件。",
        "en": "Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euler ORIGINAL signed base normalization exact quotient，不以小數近似替代。",
          "en": "Recompute Euler ORIGINAL signed base normalization exact quotient with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "23",
          "26"
        ],
        "output": "23/26",
        "label": "Euler ORIGINAL signed base normalization exact quotient"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "Euler ORIGINAL signed base normalization floor quotient multiple := mul(0, 26)",
      "after": "Euler ORIGINAL signed base normalization floor quotient multiple = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留索引、正負號、分母與證明適用條件。",
        "en": "Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euler ORIGINAL signed base normalization floor quotient multiple，不以小數近似替代。",
          "en": "Recompute Euler ORIGINAL signed base normalization floor quotient multiple with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "26"
        ],
        "output": "0",
        "label": "Euler ORIGINAL signed base normalization floor quotient multiple"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "Euler ORIGINAL signed base normalization normalized remainder := sub(23, 0)",
      "after": "Euler ORIGINAL signed base normalization normalized remainder = 23",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留索引、正負號、分母與證明適用條件。",
        "en": "Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euler ORIGINAL signed base normalization normalized remainder，不以小數近似替代。",
          "en": "Recompute Euler ORIGINAL signed base normalization normalized remainder with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "23",
          "0"
        ],
        "output": "23",
        "label": "Euler ORIGINAL signed base normalization normalized remainder"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "Euler ORIGINAL signed base normalization:ORIGINAL signed integer=23;modulus=26;exact ratio=23/26;floor quotient=0",
      "after": "value=q*n+remainder;normalized residue=23 in[0,26);same congruence class,NOT absolute-value replacement",
      "relation": "implies",
      "rule_id": "euler_exact_signed_remainder_not_abs_or_integer_equality",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對正模數取向下整數商，負base也保原同餘類；不能改成abs(a)。相同餘數是mod n同餘，不是原整數相等。",
        "en": "Use the floor quotient for a positive modulus,retaining the congruence class of a negative base rather than replacing it by abs(a).Equal residues mean congruence modulo n,not equality of original integers."
      },
      "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": "Euler ORIGINAL base applicability gcd:ORIGINAL gcd inputs=[23, 26];nonnegative initial pair=[23, 26]",
      "after": "ALL exact Euclidean rows[left,right,q,r]=[[23, 26, 0, 23], [26, 23, 1, 3], [23, 3, 7, 2], [3, 2, 1, 1], [2, 1, 2, 0]];terminal gcd=1",
      "relation": "implies",
      "rule_id": "euler_gcd_full_remainder_chain_unit_test",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每列left=q·right+r且0≤r<正right，餘數替換保共同因數集並嚴格下降，終端給gcd；包含gcd(0,n)=n，不除以0。",
        "en": "Each row has left=q*right+r with0≤r<positive right.Remainder replacement preserves common divisors and strictly descends to the gcd,including gcd(0,n)=n without division by0."
      },
      "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 residue representative0:ORIGINAL gcd inputs=[0, 26];nonnegative initial pair=[0, 26]",
      "after": "ALL exact Euclidean rows[left,right,q,r]=[[0, 26, 0, 0]];terminal gcd=26",
      "relation": "implies",
      "rule_id": "euler_gcd_full_remainder_chain_unit_test",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每列left=q·right+r且0≤r<正right，餘數替換保共同因數集並嚴格下降，終端給gcd；包含gcd(0,n)=n，不除以0。",
        "en": "Each row has left=q*right+r with0≤r<positive right.Remainder replacement preserves common divisors and strictly descends to the gcd,including gcd(0,n)=n without division by0."
      },
      "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": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "Euler residue representative1:ORIGINAL gcd inputs=[1, 26];nonnegative initial pair=[1, 26]",
      "after": "ALL exact Euclidean rows[left,right,q,r]=[[1, 26, 0, 1], [26, 1, 26, 0]];terminal gcd=1",
      "relation": "implies",
      "rule_id": "euler_gcd_full_remainder_chain_unit_test",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每列left=q·right+r且0≤r<正right，餘數替換保共同因數集並嚴格下降，終端給gcd；包含gcd(0,n)=n，不除以0。",
        "en": "Each row has left=q*right+r with0≤r<positive right.Remainder replacement preserves common divisors and strictly descends to the gcd,including gcd(0,n)=n without division by0."
      },
      "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": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "Euler residue representative2:ORIGINAL gcd inputs=[2, 26];nonnegative initial pair=[2, 26]",
      "after": "ALL exact Euclidean rows[left,right,q,r]=[[2, 26, 0, 2], [26, 2, 13, 0]];terminal gcd=2",
      "relation": "implies",
      "rule_id": "euler_gcd_full_remainder_chain_unit_test",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每列left=q·right+r且0≤r<正right，餘數替換保共同因數集並嚴格下降，終端給gcd；包含gcd(0,n)=n，不除以0。",
        "en": "Each row has left=q*right+r with0≤r<positive right.Remainder replacement preserves common divisors and strictly descends to the gcd,including gcd(0,n)=n without division by0."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "Euler residue representative3:ORIGINAL gcd inputs=[3, 26];nonnegative initial pair=[3, 26]",
      "after": "ALL exact Euclidean rows[left,right,q,r]=[[3, 26, 0, 3], [26, 3, 8, 2], [3, 2, 1, 1], [2, 1, 2, 0]];terminal gcd=1",
      "relation": "implies",
      "rule_id": "euler_gcd_full_remainder_chain_unit_test",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每列left=q·right+r且0≤r<正right，餘數替換保共同因數集並嚴格下降，終端給gcd；包含gcd(0,n)=n，不除以0。",
        "en": "Each row has left=q*right+r with0≤r<positive right.Remainder replacement preserves common divisors and strictly descends to the gcd,including gcd(0,n)=n without division by0."
      },
      "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": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "Euler residue representative4:ORIGINAL gcd inputs=[4, 26];nonnegative initial pair=[4, 26]",
      "after": "ALL exact Euclidean rows[left,right,q,r]=[[4, 26, 0, 4], [26, 4, 6, 2], [4, 2, 2, 0]];terminal gcd=2",
      "relation": "implies",
      "rule_id": "euler_gcd_full_remainder_chain_unit_test",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每列left=q·right+r且0≤r<正right，餘數替換保共同因數集並嚴格下降，終端給gcd；包含gcd(0,n)=n，不除以0。",
        "en": "Each row has left=q*right+r with0≤r<positive right.Remainder replacement preserves common divisors and strictly descends to the gcd,including gcd(0,n)=n without division by0."
      },
      "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": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "Euler residue representative5:ORIGINAL gcd inputs=[5, 26];nonnegative initial pair=[5, 26]",
      "after": "ALL exact Euclidean rows[left,right,q,r]=[[5, 26, 0, 5], [26, 5, 5, 1], [5, 1, 5, 0]];terminal gcd=1",
      "relation": "implies",
      "rule_id": "euler_gcd_full_remainder_chain_unit_test",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每列left=q·right+r且0≤r<正right，餘數替換保共同因數集並嚴格下降，終端給gcd；包含gcd(0,n)=n，不除以0。",
        "en": "Each row has left=q*right+r with0≤r<positive right.Remainder replacement preserves common divisors and strictly descends to the gcd,including gcd(0,n)=n without division by0."
      },
      "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": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "Euler residue representative6:ORIGINAL gcd inputs=[6, 26];nonnegative initial pair=[6, 26]",
      "after": "ALL exact Euclidean rows[left,right,q,r]=[[6, 26, 0, 6], [26, 6, 4, 2], [6, 2, 3, 0]];terminal gcd=2",
      "relation": "implies",
      "rule_id": "euler_gcd_full_remainder_chain_unit_test",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每列left=q·right+r且0≤r<正right，餘數替換保共同因數集並嚴格下降，終端給gcd；包含gcd(0,n)=n，不除以0。",
        "en": "Each row has left=q*right+r with0≤r<positive right.Remainder replacement preserves common divisors and strictly descends to the gcd,including gcd(0,n)=n without division by0."
      },
      "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": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "Euler residue representative7:ORIGINAL gcd inputs=[7, 26];nonnegative initial pair=[7, 26]",
      "after": "ALL exact Euclidean rows[left,right,q,r]=[[7, 26, 0, 7], [26, 7, 3, 5], [7, 5, 1, 2], [5, 2, 2, 1], [2, 1, 2, 0]];terminal gcd=1",
      "relation": "implies",
      "rule_id": "euler_gcd_full_remainder_chain_unit_test",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每列left=q·right+r且0≤r<正right，餘數替換保共同因數集並嚴格下降，終端給gcd；包含gcd(0,n)=n，不除以0。",
        "en": "Each row has left=q*right+r with0≤r<positive right.Remainder replacement preserves common divisors and strictly descends to the gcd,including gcd(0,n)=n without division by0."
      },
      "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": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "Euler residue representative8:ORIGINAL gcd inputs=[8, 26];nonnegative initial pair=[8, 26]",
      "after": "ALL exact Euclidean rows[left,right,q,r]=[[8, 26, 0, 8], [26, 8, 3, 2], [8, 2, 4, 0]];terminal gcd=2",
      "relation": "implies",
      "rule_id": "euler_gcd_full_remainder_chain_unit_test",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每列left=q·right+r且0≤r<正right，餘數替換保共同因數集並嚴格下降，終端給gcd；包含gcd(0,n)=n，不除以0。",
        "en": "Each row has left=q*right+r with0≤r<positive right.Remainder replacement preserves common divisors and strictly descends to the gcd,including gcd(0,n)=n without division by0."
      },
      "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": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "Euler residue representative9:ORIGINAL gcd inputs=[9, 26];nonnegative initial pair=[9, 26]",
      "after": "ALL exact Euclidean rows[left,right,q,r]=[[9, 26, 0, 9], [26, 9, 2, 8], [9, 8, 1, 1], [8, 1, 8, 0]];terminal gcd=1",
      "relation": "implies",
      "rule_id": "euler_gcd_full_remainder_chain_unit_test",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每列left=q·right+r且0≤r<正right，餘數替換保共同因數集並嚴格下降，終端給gcd；包含gcd(0,n)=n，不除以0。",
        "en": "Each row has left=q*right+r with0≤r<positive right.Remainder replacement preserves common divisors and strictly descends to the gcd,including gcd(0,n)=n without division by0."
      },
      "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": "S017",
      "uses": [
        "G001",
        "S016"
      ],
      "before": "Euler residue representative10:ORIGINAL gcd inputs=[10, 26];nonnegative initial pair=[10, 26]",
      "after": "ALL exact Euclidean rows[left,right,q,r]=[[10, 26, 0, 10], [26, 10, 2, 6], [10, 6, 1, 4], [6, 4, 1, 2], [4, 2, 2, 0]];terminal gcd=2",
      "relation": "implies",
      "rule_id": "euler_gcd_full_remainder_chain_unit_test",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每列left=q·right+r且0≤r<正right，餘數替換保共同因數集並嚴格下降，終端給gcd；包含gcd(0,n)=n，不除以0。",
        "en": "Each row has left=q*right+r with0≤r<positive right.Remainder replacement preserves common divisors and strictly descends to the gcd,including gcd(0,n)=n without division by0."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S018",
      "uses": [
        "G001",
        "S017"
      ],
      "before": "Euler residue representative11:ORIGINAL gcd inputs=[11, 26];nonnegative initial pair=[11, 26]",
      "after": "ALL exact Euclidean rows[left,right,q,r]=[[11, 26, 0, 11], [26, 11, 2, 4], [11, 4, 2, 3], [4, 3, 1, 1], [3, 1, 3, 0]];terminal gcd=1",
      "relation": "implies",
      "rule_id": "euler_gcd_full_remainder_chain_unit_test",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每列left=q·right+r且0≤r<正right，餘數替換保共同因數集並嚴格下降，終端給gcd；包含gcd(0,n)=n，不除以0。",
        "en": "Each row has left=q*right+r with0≤r<positive right.Remainder replacement preserves common divisors and strictly descends to the gcd,including gcd(0,n)=n without division by0."
      },
      "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": "S019",
      "uses": [
        "G001",
        "S018"
      ],
      "before": "Euler residue representative12:ORIGINAL gcd inputs=[12, 26];nonnegative initial pair=[12, 26]",
      "after": "ALL exact Euclidean rows[left,right,q,r]=[[12, 26, 0, 12], [26, 12, 2, 2], [12, 2, 6, 0]];terminal gcd=2",
      "relation": "implies",
      "rule_id": "euler_gcd_full_remainder_chain_unit_test",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每列left=q·right+r且0≤r<正right，餘數替換保共同因數集並嚴格下降，終端給gcd；包含gcd(0,n)=n，不除以0。",
        "en": "Each row has left=q*right+r with0≤r<positive right.Remainder replacement preserves common divisors and strictly descends to the gcd,including gcd(0,n)=n without division by0."
      },
      "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": "S020",
      "uses": [
        "G001",
        "S019"
      ],
      "before": "Euler residue representative13:ORIGINAL gcd inputs=[13, 26];nonnegative initial pair=[13, 26]",
      "after": "ALL exact Euclidean rows[left,right,q,r]=[[13, 26, 0, 13], [26, 13, 2, 0]];terminal gcd=13",
      "relation": "implies",
      "rule_id": "euler_gcd_full_remainder_chain_unit_test",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每列left=q·right+r且0≤r<正right，餘數替換保共同因數集並嚴格下降，終端給gcd；包含gcd(0,n)=n，不除以0。",
        "en": "Each row has left=q*right+r with0≤r<positive right.Remainder replacement preserves common divisors and strictly descends to the gcd,including gcd(0,n)=n without division by0."
      },
      "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": "S021",
      "uses": [
        "G001",
        "S020"
      ],
      "before": "Euler residue representative14:ORIGINAL gcd inputs=[14, 26];nonnegative initial pair=[14, 26]",
      "after": "ALL exact Euclidean rows[left,right,q,r]=[[14, 26, 0, 14], [26, 14, 1, 12], [14, 12, 1, 2], [12, 2, 6, 0]];terminal gcd=2",
      "relation": "implies",
      "rule_id": "euler_gcd_full_remainder_chain_unit_test",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每列left=q·right+r且0≤r<正right，餘數替換保共同因數集並嚴格下降，終端給gcd；包含gcd(0,n)=n，不除以0。",
        "en": "Each row has left=q*right+r with0≤r<positive right.Remainder replacement preserves common divisors and strictly descends to the gcd,including gcd(0,n)=n without division by0."
      },
      "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": "S022",
      "uses": [
        "G001",
        "S021"
      ],
      "before": "Euler residue representative15:ORIGINAL gcd inputs=[15, 26];nonnegative initial pair=[15, 26]",
      "after": "ALL exact Euclidean rows[left,right,q,r]=[[15, 26, 0, 15], [26, 15, 1, 11], [15, 11, 1, 4], [11, 4, 2, 3], [4, 3, 1, 1], [3, 1, 3, 0]];terminal gcd=1",
      "relation": "implies",
      "rule_id": "euler_gcd_full_remainder_chain_unit_test",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每列left=q·right+r且0≤r<正right，餘數替換保共同因數集並嚴格下降，終端給gcd；包含gcd(0,n)=n，不除以0。",
        "en": "Each row has left=q*right+r with0≤r<positive right.Remainder replacement preserves common divisors and strictly descends to the gcd,including gcd(0,n)=n without division by0."
      },
      "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": "S023",
      "uses": [
        "G001",
        "S022"
      ],
      "before": "Euler residue representative16:ORIGINAL gcd inputs=[16, 26];nonnegative initial pair=[16, 26]",
      "after": "ALL exact Euclidean rows[left,right,q,r]=[[16, 26, 0, 16], [26, 16, 1, 10], [16, 10, 1, 6], [10, 6, 1, 4], [6, 4, 1, 2], [4, 2, 2, 0]];terminal gcd=2",
      "relation": "implies",
      "rule_id": "euler_gcd_full_remainder_chain_unit_test",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每列left=q·right+r且0≤r<正right，餘數替換保共同因數集並嚴格下降，終端給gcd；包含gcd(0,n)=n，不除以0。",
        "en": "Each row has left=q*right+r with0≤r<positive right.Remainder replacement preserves common divisors and strictly descends to the gcd,including gcd(0,n)=n without division by0."
      },
      "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": "S024",
      "uses": [
        "G001",
        "S023"
      ],
      "before": "Euler residue representative17:ORIGINAL gcd inputs=[17, 26];nonnegative initial pair=[17, 26]",
      "after": "ALL exact Euclidean rows[left,right,q,r]=[[17, 26, 0, 17], [26, 17, 1, 9], [17, 9, 1, 8], [9, 8, 1, 1], [8, 1, 8, 0]];terminal gcd=1",
      "relation": "implies",
      "rule_id": "euler_gcd_full_remainder_chain_unit_test",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每列left=q·right+r且0≤r<正right，餘數替換保共同因數集並嚴格下降，終端給gcd；包含gcd(0,n)=n，不除以0。",
        "en": "Each row has left=q*right+r with0≤r<positive right.Remainder replacement preserves common divisors and strictly descends to the gcd,including gcd(0,n)=n without division by0."
      },
      "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": "S025",
      "uses": [
        "G001",
        "S024"
      ],
      "before": "Euler residue representative18:ORIGINAL gcd inputs=[18, 26];nonnegative initial pair=[18, 26]",
      "after": "ALL exact Euclidean rows[left,right,q,r]=[[18, 26, 0, 18], [26, 18, 1, 8], [18, 8, 2, 2], [8, 2, 4, 0]];terminal gcd=2",
      "relation": "implies",
      "rule_id": "euler_gcd_full_remainder_chain_unit_test",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每列left=q·right+r且0≤r<正right，餘數替換保共同因數集並嚴格下降，終端給gcd；包含gcd(0,n)=n，不除以0。",
        "en": "Each row has left=q*right+r with0≤r<positive right.Remainder replacement preserves common divisors and strictly descends to the gcd,including gcd(0,n)=n without division by0."
      },
      "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": "S026",
      "uses": [
        "G001",
        "S025"
      ],
      "before": "Euler residue representative19:ORIGINAL gcd inputs=[19, 26];nonnegative initial pair=[19, 26]",
      "after": "ALL exact Euclidean rows[left,right,q,r]=[[19, 26, 0, 19], [26, 19, 1, 7], [19, 7, 2, 5], [7, 5, 1, 2], [5, 2, 2, 1], [2, 1, 2, 0]];terminal gcd=1",
      "relation": "implies",
      "rule_id": "euler_gcd_full_remainder_chain_unit_test",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每列left=q·right+r且0≤r<正right，餘數替換保共同因數集並嚴格下降，終端給gcd；包含gcd(0,n)=n，不除以0。",
        "en": "Each row has left=q*right+r with0≤r<positive right.Remainder replacement preserves common divisors and strictly descends to the gcd,including gcd(0,n)=n without division by0."
      },
      "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": "S027",
      "uses": [
        "G001",
        "S026"
      ],
      "before": "Euler residue representative20:ORIGINAL gcd inputs=[20, 26];nonnegative initial pair=[20, 26]",
      "after": "ALL exact Euclidean rows[left,right,q,r]=[[20, 26, 0, 20], [26, 20, 1, 6], [20, 6, 3, 2], [6, 2, 3, 0]];terminal gcd=2",
      "relation": "implies",
      "rule_id": "euler_gcd_full_remainder_chain_unit_test",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每列left=q·right+r且0≤r<正right，餘數替換保共同因數集並嚴格下降，終端給gcd；包含gcd(0,n)=n，不除以0。",
        "en": "Each row has left=q*right+r with0≤r<positive right.Remainder replacement preserves common divisors and strictly descends to the gcd,including gcd(0,n)=n without division by0."
      },
      "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": "S028",
      "uses": [
        "G001",
        "S027"
      ],
      "before": "Euler residue representative21:ORIGINAL gcd inputs=[21, 26];nonnegative initial pair=[21, 26]",
      "after": "ALL exact Euclidean rows[left,right,q,r]=[[21, 26, 0, 21], [26, 21, 1, 5], [21, 5, 4, 1], [5, 1, 5, 0]];terminal gcd=1",
      "relation": "implies",
      "rule_id": "euler_gcd_full_remainder_chain_unit_test",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每列left=q·right+r且0≤r<正right，餘數替換保共同因數集並嚴格下降，終端給gcd；包含gcd(0,n)=n，不除以0。",
        "en": "Each row has left=q*right+r with0≤r<positive right.Remainder replacement preserves common divisors and strictly descends to the gcd,including gcd(0,n)=n without division by0."
      },
      "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": "S029",
      "uses": [
        "G001",
        "S028"
      ],
      "before": "Euler residue representative22:ORIGINAL gcd inputs=[22, 26];nonnegative initial pair=[22, 26]",
      "after": "ALL exact Euclidean rows[left,right,q,r]=[[22, 26, 0, 22], [26, 22, 1, 4], [22, 4, 5, 2], [4, 2, 2, 0]];terminal gcd=2",
      "relation": "implies",
      "rule_id": "euler_gcd_full_remainder_chain_unit_test",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每列left=q·right+r且0≤r<正right，餘數替換保共同因數集並嚴格下降，終端給gcd；包含gcd(0,n)=n，不除以0。",
        "en": "Each row has left=q*right+r with0≤r<positive right.Remainder replacement preserves common divisors and strictly descends to the gcd,including gcd(0,n)=n without division by0."
      },
      "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": "S030",
      "uses": [
        "G001",
        "S029"
      ],
      "before": "Euler residue representative23:ORIGINAL gcd inputs=[23, 26];nonnegative initial pair=[23, 26]",
      "after": "ALL exact Euclidean rows[left,right,q,r]=[[23, 26, 0, 23], [26, 23, 1, 3], [23, 3, 7, 2], [3, 2, 1, 1], [2, 1, 2, 0]];terminal gcd=1",
      "relation": "implies",
      "rule_id": "euler_gcd_full_remainder_chain_unit_test",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每列left=q·right+r且0≤r<正right，餘數替換保共同因數集並嚴格下降，終端給gcd；包含gcd(0,n)=n，不除以0。",
        "en": "Each row has left=q*right+r with0≤r<positive right.Remainder replacement preserves common divisors and strictly descends to the gcd,including gcd(0,n)=n without division by0."
      },
      "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": "S031",
      "uses": [
        "G001",
        "S030"
      ],
      "before": "Euler residue representative24:ORIGINAL gcd inputs=[24, 26];nonnegative initial pair=[24, 26]",
      "after": "ALL exact Euclidean rows[left,right,q,r]=[[24, 26, 0, 24], [26, 24, 1, 2], [24, 2, 12, 0]];terminal gcd=2",
      "relation": "implies",
      "rule_id": "euler_gcd_full_remainder_chain_unit_test",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每列left=q·right+r且0≤r<正right，餘數替換保共同因數集並嚴格下降，終端給gcd；包含gcd(0,n)=n，不除以0。",
        "en": "Each row has left=q*right+r with0≤r<positive right.Remainder replacement preserves common divisors and strictly descends to the gcd,including gcd(0,n)=n without division by0."
      },
      "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": "S032",
      "uses": [
        "G001",
        "S031"
      ],
      "before": "Euler residue representative25:ORIGINAL gcd inputs=[25, 26];nonnegative initial pair=[25, 26]",
      "after": "ALL exact Euclidean rows[left,right,q,r]=[[25, 26, 0, 25], [26, 25, 1, 1], [25, 1, 25, 0]];terminal gcd=1",
      "relation": "implies",
      "rule_id": "euler_gcd_full_remainder_chain_unit_test",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每列left=q·right+r且0≤r<正right，餘數替換保共同因數集並嚴格下降，終端給gcd；包含gcd(0,n)=n，不除以0。",
        "en": "Each row has left=q*right+r with0≤r<positive right.Remainder replacement preserves common divisors and strictly descends to the gcd,including gcd(0,n)=n without division by0."
      },
      "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": "S033",
      "uses": [
        "G001",
        "S032"
      ],
      "before": "Euler complete reduced residue count phi := sum(1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1)",
      "after": "Euler complete reduced residue count phi = 12",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留索引、正負號、分母與證明適用條件。",
        "en": "Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euler complete reduced residue count phi，不以小數近似替代。",
          "en": "Recompute Euler complete reduced residue count phi with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1",
          "1",
          "1",
          "1",
          "1",
          "1",
          "1",
          "1",
          "1",
          "1",
          "1",
          "1"
        ],
        "output": "12",
        "label": "Euler complete reduced residue count phi"
      }
    },
    {
      "id": "S034",
      "uses": [
        "G001",
        "S033"
      ],
      "before": "COMPLETE original residue carrier0..25;ALL [representative,gcd,unit] rows=[[0, 26, 否], [1, 1, 是], [2, 2, 否], [3, 1, 是], [4, 2, 否], [5, 1, 是], [6, 2, 否], [7, 1, 是], [8, 2, 否], [9, 1, 是], [10, 2, 否], [11, 1, 是], [12, 2, 否], [13, 13, 否], [14, 2, 否], [15, 1, 是], [16, 2, 否], [17, 1, 是], [18, 2, 否], [19, 1, 是], [20, 2, 否], [21, 1, 是], [22, 2, 否], [23, 1, 是], [24, 2, 否], [25, 1, 是]]",
      "after": "increasing COMPLETE U(n)=[1, 3, 5, 7, 9, 11, 15, 17, 19, 21, 23, 25];phi(n)=12;source count over1..n agrees because0 and n are BOTH nonunits for n>=2",
      "relation": "implies",
      "rule_id": "euler_full_units_totient_count_convention_not_prime_only",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐所有代表判gcd1而得完整單位元集合，1必在其中。來源φ用1..n而unit list用0..n−1；因n≥2，換出的0與n都非單位元，計數一致，不擅自把n當prime。",
        "en": "Test gcd1 for every representative to obtain the complete unit set,which contains1.The source totient count uses1..n while the unit list uses0..n−1.For n≥2 both swapped endpoints0 and n are nonunits,so the counts agree without assuming a prime modulus."
      },
      "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": "S035",
      "uses": [
        "G001",
        "S034"
      ],
      "before": "ORIGINAL normalized base=23;COMPLETE totient exponent=12;modulus=26",
      "after": "initialize accumulator1,current_base=base,remaining=phi;invariant accumulator*current_base^remaining congruent to the original totient power",
      "relation": "implies",
      "rule_id": "euler_totient_power_initial_exact_invariant",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以有限次平方乘法保原φ次方同餘，不建立龐大普通整數冪。φ是完整單位元數，不把它換成n−1除非另外證n為prime。",
        "en": "Use finite square-and-multiply steps preserving the original totient-power congruence,without constructing huge ordinary powers.Phi is the complete unit count,not n−1 unless primality is separately established."
      },
      "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": "S036",
      "uses": [
        "G001",
        "S035"
      ],
      "before": "Euler power bit0 base square := mul(23, 23)",
      "after": "Euler power bit0 base square = 529",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留索引、正負號、分母與證明適用條件。",
        "en": "Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euler power bit0 base square，不以小數近似替代。",
          "en": "Recompute Euler power bit0 base square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "23",
          "23"
        ],
        "output": "529",
        "label": "Euler power bit0 base square"
      }
    },
    {
      "id": "S037",
      "uses": [
        "G001",
        "S036"
      ],
      "before": "Euler power bit0 squared-base reduction exact quotient := div(529, 26)",
      "after": "Euler power bit0 squared-base reduction exact quotient = 529/26",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留索引、正負號、分母與證明適用條件。",
        "en": "Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euler power bit0 squared-base reduction exact quotient，不以小數近似替代。",
          "en": "Recompute Euler power bit0 squared-base reduction exact quotient with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "529",
          "26"
        ],
        "output": "529/26",
        "label": "Euler power bit0 squared-base reduction exact quotient"
      }
    },
    {
      "id": "S038",
      "uses": [
        "G001",
        "S037"
      ],
      "before": "Euler power bit0 squared-base reduction floor quotient multiple := mul(20, 26)",
      "after": "Euler power bit0 squared-base reduction floor quotient multiple = 520",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留索引、正負號、分母與證明適用條件。",
        "en": "Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euler power bit0 squared-base reduction floor quotient multiple，不以小數近似替代。",
          "en": "Recompute Euler power bit0 squared-base reduction floor quotient multiple with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "20",
          "26"
        ],
        "output": "520",
        "label": "Euler power bit0 squared-base reduction floor quotient multiple"
      }
    },
    {
      "id": "S039",
      "uses": [
        "G001",
        "S038"
      ],
      "before": "Euler power bit0 squared-base reduction normalized remainder := sub(529, 520)",
      "after": "Euler power bit0 squared-base reduction normalized remainder = 9",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留索引、正負號、分母與證明適用條件。",
        "en": "Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euler power bit0 squared-base reduction normalized remainder，不以小數近似替代。",
          "en": "Recompute Euler power bit0 squared-base reduction normalized remainder with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "529",
          "520"
        ],
        "output": "9",
        "label": "Euler power bit0 squared-base reduction normalized remainder"
      }
    },
    {
      "id": "S040",
      "uses": [
        "G001",
        "S039"
      ],
      "before": "Euler power bit0 squared-base reduction:ORIGINAL signed integer=529;modulus=26;exact ratio=529/26;floor quotient=20",
      "after": "value=q*n+remainder;normalized residue=9 in[0,26);same congruence class,NOT absolute-value replacement",
      "relation": "implies",
      "rule_id": "euler_exact_signed_remainder_not_abs_or_integer_equality",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對正模數取向下整數商，負base也保原同餘類；不能改成abs(a)。相同餘數是mod n同餘，不是原整數相等。",
        "en": "Use the floor quotient for a positive modulus,retaining the congruence class of a negative base rather than replacing it by abs(a).Equal residues mean congruence modulo n,not equality of original integers."
      },
      "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": "S041",
      "uses": [
        "G001",
        "S040"
      ],
      "before": "step=0;previous[accumulator,base,exponent]=[1, 23, 12];low bit=0",
      "after": "next[accumulator,base,exponent]=[1, 9, 6];split exponent=2*q+bit preserves original-power invariant",
      "relation": "implies",
      "rule_id": "euler_every_binary_power_step_unit_not_required_for_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "奇次方取一個base進accumulator，偶部分改用平方base，取模保同餘且exponent減半終止。這段精確算術對nonunit也能做；能計算不等於Euler定理前提成立。",
        "en": "For an odd power move one base factor into the accumulator and square the base for the even part.Exact reductions preserve congruence and halving guarantees termination.This arithmetic also works for nonunits;computability does not establish Euler applicability."
      },
      "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": "S042",
      "uses": [
        "G001",
        "S041"
      ],
      "before": "Euler power bit1 base square := mul(9, 9)",
      "after": "Euler power bit1 base square = 81",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留索引、正負號、分母與證明適用條件。",
        "en": "Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euler power bit1 base square，不以小數近似替代。",
          "en": "Recompute Euler power bit1 base square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "9",
          "9"
        ],
        "output": "81",
        "label": "Euler power bit1 base square"
      }
    },
    {
      "id": "S043",
      "uses": [
        "G001",
        "S042"
      ],
      "before": "Euler power bit1 squared-base reduction exact quotient := div(81, 26)",
      "after": "Euler power bit1 squared-base reduction exact quotient = 81/26",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留索引、正負號、分母與證明適用條件。",
        "en": "Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euler power bit1 squared-base reduction exact quotient，不以小數近似替代。",
          "en": "Recompute Euler power bit1 squared-base reduction exact quotient with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "81",
          "26"
        ],
        "output": "81/26",
        "label": "Euler power bit1 squared-base reduction exact quotient"
      }
    },
    {
      "id": "S044",
      "uses": [
        "G001",
        "S043"
      ],
      "before": "Euler power bit1 squared-base reduction floor quotient multiple := mul(3, 26)",
      "after": "Euler power bit1 squared-base reduction floor quotient multiple = 78",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留索引、正負號、分母與證明適用條件。",
        "en": "Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euler power bit1 squared-base reduction floor quotient multiple，不以小數近似替代。",
          "en": "Recompute Euler power bit1 squared-base reduction floor quotient multiple with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "3",
          "26"
        ],
        "output": "78",
        "label": "Euler power bit1 squared-base reduction floor quotient multiple"
      }
    },
    {
      "id": "S045",
      "uses": [
        "G001",
        "S044"
      ],
      "before": "Euler power bit1 squared-base reduction normalized remainder := sub(81, 78)",
      "after": "Euler power bit1 squared-base reduction normalized remainder = 3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留索引、正負號、分母與證明適用條件。",
        "en": "Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euler power bit1 squared-base reduction normalized remainder，不以小數近似替代。",
          "en": "Recompute Euler power bit1 squared-base reduction normalized remainder with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "81",
          "78"
        ],
        "output": "3",
        "label": "Euler power bit1 squared-base reduction normalized remainder"
      }
    },
    {
      "id": "S046",
      "uses": [
        "G001",
        "S045"
      ],
      "before": "Euler power bit1 squared-base reduction:ORIGINAL signed integer=81;modulus=26;exact ratio=81/26;floor quotient=3",
      "after": "value=q*n+remainder;normalized residue=3 in[0,26);same congruence class,NOT absolute-value replacement",
      "relation": "implies",
      "rule_id": "euler_exact_signed_remainder_not_abs_or_integer_equality",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對正模數取向下整數商，負base也保原同餘類；不能改成abs(a)。相同餘數是mod n同餘，不是原整數相等。",
        "en": "Use the floor quotient for a positive modulus,retaining the congruence class of a negative base rather than replacing it by abs(a).Equal residues mean congruence modulo n,not equality of original integers."
      },
      "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": "S047",
      "uses": [
        "G001",
        "S046"
      ],
      "before": "step=1;previous[accumulator,base,exponent]=[1, 9, 6];low bit=0",
      "after": "next[accumulator,base,exponent]=[1, 3, 3];split exponent=2*q+bit preserves original-power invariant",
      "relation": "implies",
      "rule_id": "euler_every_binary_power_step_unit_not_required_for_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "奇次方取一個base進accumulator，偶部分改用平方base，取模保同餘且exponent減半終止。這段精確算術對nonunit也能做；能計算不等於Euler定理前提成立。",
        "en": "For an odd power move one base factor into the accumulator and square the base for the even part.Exact reductions preserve congruence and halving guarantees termination.This arithmetic also works for nonunits;computability does not establish Euler applicability."
      },
      "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": "S048",
      "uses": [
        "G001",
        "S047"
      ],
      "before": "Euler power bit2 accumulator product := mul(1, 3)",
      "after": "Euler power bit2 accumulator product = 3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留索引、正負號、分母與證明適用條件。",
        "en": "Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euler power bit2 accumulator product，不以小數近似替代。",
          "en": "Recompute Euler power bit2 accumulator product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "3"
        ],
        "output": "3",
        "label": "Euler power bit2 accumulator product"
      }
    },
    {
      "id": "S049",
      "uses": [
        "G001",
        "S048"
      ],
      "before": "Euler power bit2 accumulator reduction exact quotient := div(3, 26)",
      "after": "Euler power bit2 accumulator reduction exact quotient = 3/26",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留索引、正負號、分母與證明適用條件。",
        "en": "Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euler power bit2 accumulator reduction exact quotient，不以小數近似替代。",
          "en": "Recompute Euler power bit2 accumulator reduction exact quotient with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "3",
          "26"
        ],
        "output": "3/26",
        "label": "Euler power bit2 accumulator reduction exact quotient"
      }
    },
    {
      "id": "S050",
      "uses": [
        "G001",
        "S049"
      ],
      "before": "Euler power bit2 accumulator reduction floor quotient multiple := mul(0, 26)",
      "after": "Euler power bit2 accumulator reduction floor quotient multiple = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留索引、正負號、分母與證明適用條件。",
        "en": "Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euler power bit2 accumulator reduction floor quotient multiple，不以小數近似替代。",
          "en": "Recompute Euler power bit2 accumulator reduction floor quotient multiple with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "26"
        ],
        "output": "0",
        "label": "Euler power bit2 accumulator reduction floor quotient multiple"
      }
    },
    {
      "id": "S051",
      "uses": [
        "G001",
        "S050"
      ],
      "before": "Euler power bit2 accumulator reduction normalized remainder := sub(3, 0)",
      "after": "Euler power bit2 accumulator reduction normalized remainder = 3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留索引、正負號、分母與證明適用條件。",
        "en": "Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euler power bit2 accumulator reduction normalized remainder，不以小數近似替代。",
          "en": "Recompute Euler power bit2 accumulator reduction normalized remainder with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "3",
          "0"
        ],
        "output": "3",
        "label": "Euler power bit2 accumulator reduction normalized remainder"
      }
    },
    {
      "id": "S052",
      "uses": [
        "G001",
        "S051"
      ],
      "before": "Euler power bit2 accumulator reduction:ORIGINAL signed integer=3;modulus=26;exact ratio=3/26;floor quotient=0",
      "after": "value=q*n+remainder;normalized residue=3 in[0,26);same congruence class,NOT absolute-value replacement",
      "relation": "implies",
      "rule_id": "euler_exact_signed_remainder_not_abs_or_integer_equality",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對正模數取向下整數商，負base也保原同餘類；不能改成abs(a)。相同餘數是mod n同餘，不是原整數相等。",
        "en": "Use the floor quotient for a positive modulus,retaining the congruence class of a negative base rather than replacing it by abs(a).Equal residues mean congruence modulo n,not equality of original integers."
      },
      "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": "S053",
      "uses": [
        "G001",
        "S052"
      ],
      "before": "Euler power bit2 base square := mul(3, 3)",
      "after": "Euler power bit2 base square = 9",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留索引、正負號、分母與證明適用條件。",
        "en": "Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euler power bit2 base square，不以小數近似替代。",
          "en": "Recompute Euler power bit2 base square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "3",
          "3"
        ],
        "output": "9",
        "label": "Euler power bit2 base square"
      }
    },
    {
      "id": "S054",
      "uses": [
        "G001",
        "S053"
      ],
      "before": "Euler power bit2 squared-base reduction exact quotient := div(9, 26)",
      "after": "Euler power bit2 squared-base reduction exact quotient = 9/26",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留索引、正負號、分母與證明適用條件。",
        "en": "Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euler power bit2 squared-base reduction exact quotient，不以小數近似替代。",
          "en": "Recompute Euler power bit2 squared-base reduction exact quotient with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "9",
          "26"
        ],
        "output": "9/26",
        "label": "Euler power bit2 squared-base reduction exact quotient"
      }
    },
    {
      "id": "S055",
      "uses": [
        "G001",
        "S054"
      ],
      "before": "Euler power bit2 squared-base reduction floor quotient multiple := mul(0, 26)",
      "after": "Euler power bit2 squared-base reduction floor quotient multiple = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留索引、正負號、分母與證明適用條件。",
        "en": "Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euler power bit2 squared-base reduction floor quotient multiple，不以小數近似替代。",
          "en": "Recompute Euler power bit2 squared-base reduction floor quotient multiple with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "26"
        ],
        "output": "0",
        "label": "Euler power bit2 squared-base reduction floor quotient multiple"
      }
    },
    {
      "id": "S056",
      "uses": [
        "G001",
        "S055"
      ],
      "before": "Euler power bit2 squared-base reduction normalized remainder := sub(9, 0)",
      "after": "Euler power bit2 squared-base reduction normalized remainder = 9",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留索引、正負號、分母與證明適用條件。",
        "en": "Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euler power bit2 squared-base reduction normalized remainder，不以小數近似替代。",
          "en": "Recompute Euler power bit2 squared-base reduction normalized remainder with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "9",
          "0"
        ],
        "output": "9",
        "label": "Euler power bit2 squared-base reduction normalized remainder"
      }
    },
    {
      "id": "S057",
      "uses": [
        "G001",
        "S056"
      ],
      "before": "Euler power bit2 squared-base reduction:ORIGINAL signed integer=9;modulus=26;exact ratio=9/26;floor quotient=0",
      "after": "value=q*n+remainder;normalized residue=9 in[0,26);same congruence class,NOT absolute-value replacement",
      "relation": "implies",
      "rule_id": "euler_exact_signed_remainder_not_abs_or_integer_equality",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對正模數取向下整數商，負base也保原同餘類；不能改成abs(a)。相同餘數是mod n同餘，不是原整數相等。",
        "en": "Use the floor quotient for a positive modulus,retaining the congruence class of a negative base rather than replacing it by abs(a).Equal residues mean congruence modulo n,not equality of original integers."
      },
      "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": "S058",
      "uses": [
        "G001",
        "S057"
      ],
      "before": "step=2;previous[accumulator,base,exponent]=[1, 3, 3];low bit=1",
      "after": "next[accumulator,base,exponent]=[3, 9, 1];split exponent=2*q+bit preserves original-power invariant",
      "relation": "implies",
      "rule_id": "euler_every_binary_power_step_unit_not_required_for_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "奇次方取一個base進accumulator，偶部分改用平方base，取模保同餘且exponent減半終止。這段精確算術對nonunit也能做；能計算不等於Euler定理前提成立。",
        "en": "For an odd power move one base factor into the accumulator and square the base for the even part.Exact reductions preserve congruence and halving guarantees termination.This arithmetic also works for nonunits;computability does not establish Euler applicability."
      },
      "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": "S059",
      "uses": [
        "G001",
        "S058"
      ],
      "before": "Euler power bit3 accumulator product := mul(3, 9)",
      "after": "Euler power bit3 accumulator product = 27",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留索引、正負號、分母與證明適用條件。",
        "en": "Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euler power bit3 accumulator product，不以小數近似替代。",
          "en": "Recompute Euler power bit3 accumulator product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "3",
          "9"
        ],
        "output": "27",
        "label": "Euler power bit3 accumulator product"
      }
    },
    {
      "id": "S060",
      "uses": [
        "G001",
        "S059"
      ],
      "before": "Euler power bit3 accumulator reduction exact quotient := div(27, 26)",
      "after": "Euler power bit3 accumulator reduction exact quotient = 27/26",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留索引、正負號、分母與證明適用條件。",
        "en": "Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euler power bit3 accumulator reduction exact quotient，不以小數近似替代。",
          "en": "Recompute Euler power bit3 accumulator reduction exact quotient with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "27",
          "26"
        ],
        "output": "27/26",
        "label": "Euler power bit3 accumulator reduction exact quotient"
      }
    },
    {
      "id": "S061",
      "uses": [
        "G001",
        "S060"
      ],
      "before": "Euler power bit3 accumulator reduction floor quotient multiple := mul(1, 26)",
      "after": "Euler power bit3 accumulator reduction floor quotient multiple = 26",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留索引、正負號、分母與證明適用條件。",
        "en": "Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euler power bit3 accumulator reduction floor quotient multiple，不以小數近似替代。",
          "en": "Recompute Euler power bit3 accumulator reduction floor quotient multiple with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "26"
        ],
        "output": "26",
        "label": "Euler power bit3 accumulator reduction floor quotient multiple"
      }
    },
    {
      "id": "S062",
      "uses": [
        "G001",
        "S061"
      ],
      "before": "Euler power bit3 accumulator reduction normalized remainder := sub(27, 26)",
      "after": "Euler power bit3 accumulator reduction normalized remainder = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留索引、正負號、分母與證明適用條件。",
        "en": "Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euler power bit3 accumulator reduction normalized remainder，不以小數近似替代。",
          "en": "Recompute Euler power bit3 accumulator reduction normalized remainder with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "27",
          "26"
        ],
        "output": "1",
        "label": "Euler power bit3 accumulator reduction normalized remainder"
      }
    },
    {
      "id": "S063",
      "uses": [
        "G001",
        "S062"
      ],
      "before": "Euler power bit3 accumulator reduction:ORIGINAL signed integer=27;modulus=26;exact ratio=27/26;floor quotient=1",
      "after": "value=q*n+remainder;normalized residue=1 in[0,26);same congruence class,NOT absolute-value replacement",
      "relation": "implies",
      "rule_id": "euler_exact_signed_remainder_not_abs_or_integer_equality",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對正模數取向下整數商，負base也保原同餘類；不能改成abs(a)。相同餘數是mod n同餘，不是原整數相等。",
        "en": "Use the floor quotient for a positive modulus,retaining the congruence class of a negative base rather than replacing it by abs(a).Equal residues mean congruence modulo n,not equality of original integers."
      },
      "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": "S064",
      "uses": [
        "G001",
        "S063"
      ],
      "before": "Euler power bit3 base square := mul(9, 9)",
      "after": "Euler power bit3 base square = 81",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留索引、正負號、分母與證明適用條件。",
        "en": "Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euler power bit3 base square，不以小數近似替代。",
          "en": "Recompute Euler power bit3 base square with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "9",
          "9"
        ],
        "output": "81",
        "label": "Euler power bit3 base square"
      }
    },
    {
      "id": "S065",
      "uses": [
        "G001",
        "S064"
      ],
      "before": "Euler power bit3 squared-base reduction exact quotient := div(81, 26)",
      "after": "Euler power bit3 squared-base reduction exact quotient = 81/26",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留索引、正負號、分母與證明適用條件。",
        "en": "Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euler power bit3 squared-base reduction exact quotient，不以小數近似替代。",
          "en": "Recompute Euler power bit3 squared-base reduction exact quotient with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "81",
          "26"
        ],
        "output": "81/26",
        "label": "Euler power bit3 squared-base reduction exact quotient"
      }
    },
    {
      "id": "S066",
      "uses": [
        "G001",
        "S065"
      ],
      "before": "Euler power bit3 squared-base reduction floor quotient multiple := mul(3, 26)",
      "after": "Euler power bit3 squared-base reduction floor quotient multiple = 78",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留索引、正負號、分母與證明適用條件。",
        "en": "Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euler power bit3 squared-base reduction floor quotient multiple，不以小數近似替代。",
          "en": "Recompute Euler power bit3 squared-base reduction floor quotient multiple with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "3",
          "26"
        ],
        "output": "78",
        "label": "Euler power bit3 squared-base reduction floor quotient multiple"
      }
    },
    {
      "id": "S067",
      "uses": [
        "G001",
        "S066"
      ],
      "before": "Euler power bit3 squared-base reduction normalized remainder := sub(81, 78)",
      "after": "Euler power bit3 squared-base reduction normalized remainder = 3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留索引、正負號、分母與證明適用條件。",
        "en": "Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euler power bit3 squared-base reduction normalized remainder，不以小數近似替代。",
          "en": "Recompute Euler power bit3 squared-base reduction normalized remainder with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "81",
          "78"
        ],
        "output": "3",
        "label": "Euler power bit3 squared-base reduction normalized remainder"
      }
    },
    {
      "id": "S068",
      "uses": [
        "G001",
        "S067"
      ],
      "before": "Euler power bit3 squared-base reduction:ORIGINAL signed integer=81;modulus=26;exact ratio=81/26;floor quotient=3",
      "after": "value=q*n+remainder;normalized residue=3 in[0,26);same congruence class,NOT absolute-value replacement",
      "relation": "implies",
      "rule_id": "euler_exact_signed_remainder_not_abs_or_integer_equality",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對正模數取向下整數商，負base也保原同餘類；不能改成abs(a)。相同餘數是mod n同餘，不是原整數相等。",
        "en": "Use the floor quotient for a positive modulus,retaining the congruence class of a negative base rather than replacing it by abs(a).Equal residues mean congruence modulo n,not equality of original integers."
      },
      "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": "step=3;previous[accumulator,base,exponent]=[3, 9, 1];low bit=1",
      "after": "next[accumulator,base,exponent]=[1, 3, 0];split exponent=2*q+bit preserves original-power invariant",
      "relation": "implies",
      "rule_id": "euler_every_binary_power_step_unit_not_required_for_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "奇次方取一個base進accumulator，偶部分改用平方base，取模保同餘且exponent減半終止。這段精確算術對nonunit也能做；能計算不等於Euler定理前提成立。",
        "en": "For an odd power move one base factor into the accumulator and square the base for the even part.Exact reductions preserve congruence and halving guarantees termination.This arithmetic also works for nonunits;computability does not establish Euler applicability."
      },
      "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": "S070",
      "uses": [
        "G001",
        "S069"
      ],
      "before": "remaining exponent0;final normalized accumulator=1",
      "after": "EXACT original normalized base^12 modulo26=1",
      "relation": "implies",
      "rule_id": "euler_totient_power_terminal_congruence_not_primality",
      "conditions": [],
      "explanation": {
        "zh-Hant": "不變式終止後給原φ次方餘數；這是指定modulus的算術結果，不是primality certificate。",
        "en": "The terminal invariant gives the original totient-power residue.This is arithmetic modulo the stated modulus,not a primality certificate."
      },
      "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 ACTUAL base-unit inverse product := mul(23, 17)",
      "after": "Euler ACTUAL base-unit inverse product = 391",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留索引、正負號、分母與證明適用條件。",
        "en": "Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euler ACTUAL base-unit inverse product，不以小數近似替代。",
          "en": "Recompute Euler ACTUAL base-unit inverse product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "23",
          "17"
        ],
        "output": "391",
        "label": "Euler ACTUAL base-unit inverse product"
      }
    },
    {
      "id": "S072",
      "uses": [
        "G001",
        "S071"
      ],
      "before": "Euler ACTUAL inverse check exact quotient := div(391, 26)",
      "after": "Euler ACTUAL inverse check exact quotient = 391/26",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留索引、正負號、分母與證明適用條件。",
        "en": "Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euler ACTUAL inverse check exact quotient，不以小數近似替代。",
          "en": "Recompute Euler ACTUAL inverse check exact quotient with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "391",
          "26"
        ],
        "output": "391/26",
        "label": "Euler ACTUAL inverse check exact quotient"
      }
    },
    {
      "id": "S073",
      "uses": [
        "G001",
        "S072"
      ],
      "before": "Euler ACTUAL inverse check floor quotient multiple := mul(15, 26)",
      "after": "Euler ACTUAL inverse check floor quotient multiple = 390",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留索引、正負號、分母與證明適用條件。",
        "en": "Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euler ACTUAL inverse check floor quotient multiple，不以小數近似替代。",
          "en": "Recompute Euler ACTUAL inverse check floor quotient multiple with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "15",
          "26"
        ],
        "output": "390",
        "label": "Euler ACTUAL inverse check floor quotient multiple"
      }
    },
    {
      "id": "S074",
      "uses": [
        "G001",
        "S073"
      ],
      "before": "Euler ACTUAL inverse check normalized remainder := sub(391, 390)",
      "after": "Euler ACTUAL inverse check normalized remainder = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已核中間量精確代值，保留索引、正負號、分母與證明適用條件。",
        "en": "Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Euler ACTUAL inverse check normalized remainder，不以小數近似替代。",
          "en": "Recompute Euler ACTUAL inverse check normalized remainder with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "391",
          "390"
        ],
        "output": "1",
        "label": "Euler ACTUAL inverse check normalized remainder"
      }
    },
    {
      "id": "S075",
      "uses": [
        "G001",
        "S074"
      ],
      "before": "Euler ACTUAL inverse check:ORIGINAL signed integer=391;modulus=26;exact ratio=391/26;floor quotient=15",
      "after": "value=q*n+remainder;normalized residue=1 in[0,26);same congruence class,NOT absolute-value replacement",
      "relation": "implies",
      "rule_id": "euler_exact_signed_remainder_not_abs_or_integer_equality",
      "conditions": [],
      "explanation": {
        "zh-Hant": "對正模數取向下整數商，負base也保原同餘類；不能改成abs(a)。相同餘數是mod n同餘，不是原整數相等。",
        "en": "Use the floor quotient for a positive modulus,retaining the congruence class of a negative base rather than replacing it by abs(a).Equal residues mean congruence modulo n,not equality of original integers."
      },
      "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": "S076",
      "uses": [
        "G001",
        "S075"
      ],
      "before": "ACTUAL modulus n=26;gcd(original a=23,n)=1;normalized base=23;ACTUAL inverse=17;COMPLETE U(n)=[1, 3, 5, 7, 9, 11, 15, 17, 19, 21, 23, 25]",
      "after": "for ANY r in U(n),a*r remains a unit;actual inverse makes multiplication injective;finite self-injection on U(n) is surjective,so a*U(n)=U(n)",
      "relation": "implies",
      "rule_id": "euler_all_unit_residue_permutation_composite_allowed",
      "conditions": [],
      "explanation": {
        "zh-Hant": "單位元乘積的inverse可由反序inverse相乘給出，故乘a留U(n)；用a的真inverse消去得單射，有限同載體推出滿射。n可以是合數；不能把全部非零residue當單位元。",
        "en": "The reverse product of inverses inverts a product of units,so multiplication by a stays in U(n).Its actual inverse proves injectivity and a finite self-injection is surjective.The modulus may be composite;not every nonzero residue must be a unit."
      },
      "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": "SAME complete U(n)=[1, 3, 5, 7, 9, 11, 15, 17, 19, 21, 23, 25] of size phi=12;multiplication by original a=23 permutes it",
      "after": "multiply ALL unit images:a^(12)*product(U) congruent product(U);EVERY U factor is a UNIT,so product(U) has reverse-product inverse;cancel ONLY this unit to get a^phi congruent1 modulo26",
      "relation": "implies",
      "rule_id": "euler_complete_unit_product_cancellation_not_nonzero_factorial",
      "conditions": [],
      "explanation": {
        "zh-Hant": "有限乘積是同一unit集合的排列，故相等；prod(U)是unit才可消去。這不是對合數模數的所有非零元素取消，也不是對0取消；沒有prime前提，Euler congruence通過不給prime converse。",
        "en": "Products over the same permuted finite unit set agree.Cancel product(U) only because it is a unit,not because it is merely nonzero.This is not cancellation of all nonzero residues under a composite modulus or cancellation of0.No prime premise is used,and a passing Euler congruence supplies no primality converse."
      },
      "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": [
      26,
      23,
      12,
      1,
      true
    ]
  },
  "conclusion": {
    "canonical_value": "(26, 23, 12, 1, 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-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations",
      "batch_path": "public/data/questions/batches/batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations/questions.jsonl",
      "batch_sha256": "303978dffa4e2a37a6ed6de51e069dd9f193b7cfd7bf4c7dce17cd7f77217198",
      "original_line": 22,
      "original_line_sha256": "613fd545f4fc85663ec87f554f208052726c28bbcd6924fcf9ac43406960b99b",
      "question_compact_sha256": "c42477534741f16a7e77546618c67f071825ad0c35f1cdc365b2867f1c6b97b9",
      "english_row_compact_sha256": "41cdb983d50edee1c3315c349da804a4b5679da4f955590035f2d74506202197",
      "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-euler-theorem-number-theory": {
        "sha256": "d14018a9950716ffce9bc6bf1d38d0e4ef8a78193dee493ebaa0d0dda1a573b6",
        "english_sha256": "4a4904aebdd6ce063f9ed8f56abef3819408119eb8e3af341233be6c26c5b990",
        "original_path": "public/data/mko/euler-theorem-number-theory.json",
        "english_path": "public/data/i18n/editions/mko-euler-theorem-number-theory.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "521929dfe9b0919224c38312ad78e5fa56457fc886faaa12b137edd83962251e",
    "original_candidate_sha256": "bdb5e6a9a815ff0e87128b3470a6b6742abf928520a671614827cf1fe275d5a0",
    "original_candidate_line_sha256": "bdb5e6a9a815ff0e87128b3470a6b6742abf928520a671614827cf1fe275d5a0",
    "mko_binding_revision": {
      "id": "edition-symbol-bindings-20261101",
      "path": "content/edition-symbol-bindings-20261101/revision.json",
      "sha256": "e9b4c69f2c4732074c37abb624c8ab8a70afc3645c12bf9df2268ef650ad7b6c",
      "previous_english_sha256": {
        "mko-euler-theorem-number-theory": "8739010277631529ae1713b33d4b0b2edb4c201055b99f8a170cd5601cb2f7be"
      }
    }
  },
  "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-20261026-008-p40",
    "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": "6e11e35fafe32f16900ce9b522d01acb7b2ee6b463ab179392076cfbe1af8034",
      "question_first_witness_sha256": "cc0fa361fc25151729e70264340d2eae94f179f6f2a2edfe80ba5704fbbd0682"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 77,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "ee9ab9fa0ea761ec921f2d992302ea4f65fff573a32ba66d354cce391097a37a",
      "arithmetic_steps": 32
    },
    "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."
  }
}