{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261014-advance-20261107-r06-euler-theorem-number-theory-0038",
  "pilot_number": "R0039-20088",
  "number": 20088,
  "family_id": "qf-mod38-eulerthm-unit-permutation",
  "title": {
    "zh-Hant": "第 20088 題：Euler定理",
    "en": "Question 20088: Multiplication by a Unit Permutes the Complete Reduced Residues"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261014-advance-20261107-r06-euler-theorem-number-theory-0038",
    "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-unit-permutation；資料={\"n\":17,\"a\":14}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "([1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], 是, multiplication_by_a_permutes_reduced_residues)",
      "explanation_zh": "Wave6根層核對=([1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], 是, multiplication_by_a_permutes_reduced_residues)；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": 2,
      "skills": [
        "Euler定理"
      ]
    },
    "variant": {
      "family_id": "qf-mod38-eulerthm-unit-permutation",
      "seed": 202611076038,
      "parameters": {
        "n": 17,
        "a": 14,
        "domain": "Exact_integer_Euler_theorem",
        "operation": "eulerthm-unit-permutation",
        "semantic_template": "qf-mod38-eulerthm-unit-permutation/{\"a\":14,\"n\":17}"
      }
    },
    "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-0038",
    "batch_id": "batch-20261014-advance-20261107-r06-euler-theorem-number-theory-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "40bffe11fd1e3ac7fd81121eb15a853e529a769d13802c82dabd12e5bb994fee",
      "family_id": "qf-mod38-eulerthm-unit-permutation",
      "seed": 202611076038,
      "semantic_template": "qf-mod38-eulerthm-unit-permutation/{\"a\":14,\"n\":17}"
    },
    "fields": {
      "title": "Multiplication by a Unit Permutes the Complete Reduced Residues",
      "stem": "Use integer modulus n=17≥2 and original integer base a=14. 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 (units, sorted_mapped_units, permutation, certificate).",
      "answer": {
        "kind": "expression",
        "value": "([1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], Yes, multiplication_by_a_permutes_reduced_residues)",
        "explanation": "List all unit residues increasingly. Multiply each by a modulo n and sort the mapped list before comparison. Equal full lists certify a permutation; the displayed mapped list is sorted, not necessarily in the original image order. The exact result, retaining the requested order, is ([1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], Yes, multiplication_by_a_permutes_reduced_residues)."
      },
      "skills": [
        "Multiplication by a Unit Permutes the Complete Reduced Residues"
      ],
      "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-unit-permutation；資料={\"n\":17,\"a\":14}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
        "en": "Use integer modulus n=17≥2 and original integer base a=14. 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 (units, sorted_mapped_units, permutation, certificate)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source operation=eulerthm-unit-permutation;original snapshot and mathematics retained",
      "after": "explicit ordered source output fields=[units, sorted_mapped_units, permutation, certificate]",
      "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(14, 17)",
      "after": "Euler ORIGINAL signed base normalization exact quotient = 14/17",
      "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": [
          "14",
          "17"
        ],
        "output": "14/17",
        "label": "Euler ORIGINAL signed base normalization exact quotient"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "Euler ORIGINAL signed base normalization floor quotient multiple := mul(0, 17)",
      "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",
          "17"
        ],
        "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(14, 0)",
      "after": "Euler ORIGINAL signed base normalization normalized remainder = 14",
      "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": [
          "14",
          "0"
        ],
        "output": "14",
        "label": "Euler ORIGINAL signed base normalization normalized remainder"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "Euler ORIGINAL signed base normalization:ORIGINAL signed integer=14;modulus=17;exact ratio=14/17;floor quotient=0",
      "after": "value=q*n+remainder;normalized residue=14 in[0,17);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=[14, 17];nonnegative initial pair=[14, 17]",
      "after": "ALL exact Euclidean rows[left,right,q,r]=[[14, 17, 0, 14], [17, 14, 1, 3], [14, 3, 4, 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, 17];nonnegative initial pair=[0, 17]",
      "after": "ALL exact Euclidean rows[left,right,q,r]=[[0, 17, 0, 0]];terminal gcd=17",
      "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, 17];nonnegative initial pair=[1, 17]",
      "after": "ALL exact Euclidean rows[left,right,q,r]=[[1, 17, 0, 1], [17, 1, 17, 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, 17];nonnegative initial pair=[2, 17]",
      "after": "ALL exact Euclidean rows[left,right,q,r]=[[2, 17, 0, 2], [17, 2, 8, 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": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "Euler residue representative3:ORIGINAL gcd inputs=[3, 17];nonnegative initial pair=[3, 17]",
      "after": "ALL exact Euclidean rows[left,right,q,r]=[[3, 17, 0, 3], [17, 3, 5, 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, 17];nonnegative initial pair=[4, 17]",
      "after": "ALL exact Euclidean rows[left,right,q,r]=[[4, 17, 0, 4], [17, 4, 4, 1], [4, 1, 4, 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": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "Euler residue representative5:ORIGINAL gcd inputs=[5, 17];nonnegative initial pair=[5, 17]",
      "after": "ALL exact Euclidean rows[left,right,q,r]=[[5, 17, 0, 5], [17, 5, 3, 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": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "Euler residue representative6:ORIGINAL gcd inputs=[6, 17];nonnegative initial pair=[6, 17]",
      "after": "ALL exact Euclidean rows[left,right,q,r]=[[6, 17, 0, 6], [17, 6, 2, 5], [6, 5, 1, 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": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "Euler residue representative7:ORIGINAL gcd inputs=[7, 17];nonnegative initial pair=[7, 17]",
      "after": "ALL exact Euclidean rows[left,right,q,r]=[[7, 17, 0, 7], [17, 7, 2, 3], [7, 3, 2, 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": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "Euler residue representative8:ORIGINAL gcd inputs=[8, 17];nonnegative initial pair=[8, 17]",
      "after": "ALL exact Euclidean rows[left,right,q,r]=[[8, 17, 0, 8], [17, 8, 2, 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": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "Euler residue representative9:ORIGINAL gcd inputs=[9, 17];nonnegative initial pair=[9, 17]",
      "after": "ALL exact Euclidean rows[left,right,q,r]=[[9, 17, 0, 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": "S017",
      "uses": [
        "G001",
        "S016"
      ],
      "before": "Euler residue representative10:ORIGINAL gcd inputs=[10, 17];nonnegative initial pair=[10, 17]",
      "after": "ALL exact Euclidean rows[left,right,q,r]=[[10, 17, 0, 10], [17, 10, 1, 7], [10, 7, 1, 3], [7, 3, 2, 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": "S018",
      "uses": [
        "G001",
        "S017"
      ],
      "before": "Euler residue representative11:ORIGINAL gcd inputs=[11, 17];nonnegative initial pair=[11, 17]",
      "after": "ALL exact Euclidean rows[left,right,q,r]=[[11, 17, 0, 11], [17, 11, 1, 6], [11, 6, 1, 5], [6, 5, 1, 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": "S019",
      "uses": [
        "G001",
        "S018"
      ],
      "before": "Euler residue representative12:ORIGINAL gcd inputs=[12, 17];nonnegative initial pair=[12, 17]",
      "after": "ALL exact Euclidean rows[left,right,q,r]=[[12, 17, 0, 12], [17, 12, 1, 5], [12, 5, 2, 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": "S020",
      "uses": [
        "G001",
        "S019"
      ],
      "before": "Euler residue representative13:ORIGINAL gcd inputs=[13, 17];nonnegative initial pair=[13, 17]",
      "after": "ALL exact Euclidean rows[left,right,q,r]=[[13, 17, 0, 13], [17, 13, 1, 4], [13, 4, 3, 1], [4, 1, 4, 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": "S021",
      "uses": [
        "G001",
        "S020"
      ],
      "before": "Euler residue representative14:ORIGINAL gcd inputs=[14, 17];nonnegative initial pair=[14, 17]",
      "after": "ALL exact Euclidean rows[left,right,q,r]=[[14, 17, 0, 14], [17, 14, 1, 3], [14, 3, 4, 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": "S022",
      "uses": [
        "G001",
        "S021"
      ],
      "before": "Euler residue representative15:ORIGINAL gcd inputs=[15, 17];nonnegative initial pair=[15, 17]",
      "after": "ALL exact Euclidean rows[left,right,q,r]=[[15, 17, 0, 15], [17, 15, 1, 2], [15, 2, 7, 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": "S023",
      "uses": [
        "G001",
        "S022"
      ],
      "before": "Euler residue representative16:ORIGINAL gcd inputs=[16, 17];nonnegative initial pair=[16, 17]",
      "after": "ALL exact Euclidean rows[left,right,q,r]=[[16, 17, 0, 16], [17, 16, 1, 1], [16, 1, 16, 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": "S024",
      "uses": [
        "G001",
        "S023"
      ],
      "before": "Euler complete reduced residue count phi := sum(1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1)",
      "after": "Euler complete reduced residue count phi = 16",
      "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",
          "1",
          "1",
          "1",
          "1"
        ],
        "output": "16",
        "label": "Euler complete reduced residue count phi"
      }
    },
    {
      "id": "S025",
      "uses": [
        "G001",
        "S024"
      ],
      "before": "COMPLETE original residue carrier0..16;ALL [representative,gcd,unit] rows=[[0, 17, 否], [1, 1, 是], [2, 1, 是], [3, 1, 是], [4, 1, 是], [5, 1, 是], [6, 1, 是], [7, 1, 是], [8, 1, 是], [9, 1, 是], [10, 1, 是], [11, 1, 是], [12, 1, 是], [13, 1, 是], [14, 1, 是], [15, 1, 是], [16, 1, 是]]",
      "after": "increasing COMPLETE U(n)=[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16];phi(n)=16;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": "S026",
      "uses": [
        "G001",
        "S025"
      ],
      "before": "Euler ACTUAL base-unit inverse product := mul(14, 11)",
      "after": "Euler ACTUAL base-unit inverse product = 154",
      "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": [
          "14",
          "11"
        ],
        "output": "154",
        "label": "Euler ACTUAL base-unit inverse product"
      }
    },
    {
      "id": "S027",
      "uses": [
        "G001",
        "S026"
      ],
      "before": "Euler ACTUAL inverse check exact quotient := div(154, 17)",
      "after": "Euler ACTUAL inverse check exact quotient = 154/17",
      "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": [
          "154",
          "17"
        ],
        "output": "154/17",
        "label": "Euler ACTUAL inverse check exact quotient"
      }
    },
    {
      "id": "S028",
      "uses": [
        "G001",
        "S027"
      ],
      "before": "Euler ACTUAL inverse check floor quotient multiple := mul(9, 17)",
      "after": "Euler ACTUAL inverse check floor quotient multiple = 153",
      "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": [
          "9",
          "17"
        ],
        "output": "153",
        "label": "Euler ACTUAL inverse check floor quotient multiple"
      }
    },
    {
      "id": "S029",
      "uses": [
        "G001",
        "S028"
      ],
      "before": "Euler ACTUAL inverse check normalized remainder := sub(154, 153)",
      "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": [
          "154",
          "153"
        ],
        "output": "1",
        "label": "Euler ACTUAL inverse check normalized remainder"
      }
    },
    {
      "id": "S030",
      "uses": [
        "G001",
        "S029"
      ],
      "before": "Euler ACTUAL inverse check:ORIGINAL signed integer=154;modulus=17;exact ratio=154/17;floor quotient=9",
      "after": "value=q*n+remainder;normalized residue=1 in[0,17);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": "S031",
      "uses": [
        "G001",
        "S030"
      ],
      "before": "ACTUAL modulus n=17;gcd(original a=14,n)=1;normalized base=14;ACTUAL inverse=11;COMPLETE U(n)=[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]",
      "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": "S032",
      "uses": [
        "G001",
        "S031"
      ],
      "before": "Euler ORIGINAL a times unit representative1 := mul(14, 1)",
      "after": "Euler ORIGINAL a times unit representative1 = 14",
      "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 a times unit representative1，不以小數近似替代。",
          "en": "Recompute Euler ORIGINAL a times unit representative1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "14",
          "1"
        ],
        "output": "14",
        "label": "Euler ORIGINAL a times unit representative1"
      }
    },
    {
      "id": "S033",
      "uses": [
        "G001",
        "S032"
      ],
      "before": "Euler unit representative1 raw image exact quotient := div(14, 17)",
      "after": "Euler unit representative1 raw image exact quotient = 14/17",
      "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 unit representative1 raw image exact quotient，不以小數近似替代。",
          "en": "Recompute Euler unit representative1 raw image exact quotient with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "14",
          "17"
        ],
        "output": "14/17",
        "label": "Euler unit representative1 raw image exact quotient"
      }
    },
    {
      "id": "S034",
      "uses": [
        "G001",
        "S033"
      ],
      "before": "Euler unit representative1 raw image floor quotient multiple := mul(0, 17)",
      "after": "Euler unit representative1 raw image 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 unit representative1 raw image floor quotient multiple，不以小數近似替代。",
          "en": "Recompute Euler unit representative1 raw image floor quotient multiple with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "17"
        ],
        "output": "0",
        "label": "Euler unit representative1 raw image floor quotient multiple"
      }
    },
    {
      "id": "S035",
      "uses": [
        "G001",
        "S034"
      ],
      "before": "Euler unit representative1 raw image normalized remainder := sub(14, 0)",
      "after": "Euler unit representative1 raw image normalized remainder = 14",
      "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 unit representative1 raw image normalized remainder，不以小數近似替代。",
          "en": "Recompute Euler unit representative1 raw image normalized remainder with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "14",
          "0"
        ],
        "output": "14",
        "label": "Euler unit representative1 raw image normalized remainder"
      }
    },
    {
      "id": "S036",
      "uses": [
        "G001",
        "S035"
      ],
      "before": "Euler unit representative1 raw image:ORIGINAL signed integer=14;modulus=17;exact ratio=14/17;floor quotient=0",
      "after": "value=q*n+remainder;normalized residue=14 in[0,17);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": "S037",
      "uses": [
        "G001",
        "S036"
      ],
      "before": "Euler ORIGINAL a times unit representative2 := mul(14, 2)",
      "after": "Euler ORIGINAL a times unit representative2 = 28",
      "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 a times unit representative2，不以小數近似替代。",
          "en": "Recompute Euler ORIGINAL a times unit representative2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "14",
          "2"
        ],
        "output": "28",
        "label": "Euler ORIGINAL a times unit representative2"
      }
    },
    {
      "id": "S038",
      "uses": [
        "G001",
        "S037"
      ],
      "before": "Euler unit representative2 raw image exact quotient := div(28, 17)",
      "after": "Euler unit representative2 raw image exact quotient = 28/17",
      "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 unit representative2 raw image exact quotient，不以小數近似替代。",
          "en": "Recompute Euler unit representative2 raw image exact quotient with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "28",
          "17"
        ],
        "output": "28/17",
        "label": "Euler unit representative2 raw image exact quotient"
      }
    },
    {
      "id": "S039",
      "uses": [
        "G001",
        "S038"
      ],
      "before": "Euler unit representative2 raw image floor quotient multiple := mul(1, 17)",
      "after": "Euler unit representative2 raw image floor quotient multiple = 17",
      "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 unit representative2 raw image floor quotient multiple，不以小數近似替代。",
          "en": "Recompute Euler unit representative2 raw image floor quotient multiple with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "17"
        ],
        "output": "17",
        "label": "Euler unit representative2 raw image floor quotient multiple"
      }
    },
    {
      "id": "S040",
      "uses": [
        "G001",
        "S039"
      ],
      "before": "Euler unit representative2 raw image normalized remainder := sub(28, 17)",
      "after": "Euler unit representative2 raw image normalized remainder = 11",
      "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 unit representative2 raw image normalized remainder，不以小數近似替代。",
          "en": "Recompute Euler unit representative2 raw image normalized remainder with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "28",
          "17"
        ],
        "output": "11",
        "label": "Euler unit representative2 raw image normalized remainder"
      }
    },
    {
      "id": "S041",
      "uses": [
        "G001",
        "S040"
      ],
      "before": "Euler unit representative2 raw image:ORIGINAL signed integer=28;modulus=17;exact ratio=28/17;floor quotient=1",
      "after": "value=q*n+remainder;normalized residue=11 in[0,17);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": "S042",
      "uses": [
        "G001",
        "S041"
      ],
      "before": "Euler ORIGINAL a times unit representative3 := mul(14, 3)",
      "after": "Euler ORIGINAL a times unit representative3 = 42",
      "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 a times unit representative3，不以小數近似替代。",
          "en": "Recompute Euler ORIGINAL a times unit representative3 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "14",
          "3"
        ],
        "output": "42",
        "label": "Euler ORIGINAL a times unit representative3"
      }
    },
    {
      "id": "S043",
      "uses": [
        "G001",
        "S042"
      ],
      "before": "Euler unit representative3 raw image exact quotient := div(42, 17)",
      "after": "Euler unit representative3 raw image exact quotient = 42/17",
      "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 unit representative3 raw image exact quotient，不以小數近似替代。",
          "en": "Recompute Euler unit representative3 raw image exact quotient with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "42",
          "17"
        ],
        "output": "42/17",
        "label": "Euler unit representative3 raw image exact quotient"
      }
    },
    {
      "id": "S044",
      "uses": [
        "G001",
        "S043"
      ],
      "before": "Euler unit representative3 raw image floor quotient multiple := mul(2, 17)",
      "after": "Euler unit representative3 raw image floor quotient multiple = 34",
      "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 unit representative3 raw image floor quotient multiple，不以小數近似替代。",
          "en": "Recompute Euler unit representative3 raw image floor quotient multiple with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "17"
        ],
        "output": "34",
        "label": "Euler unit representative3 raw image floor quotient multiple"
      }
    },
    {
      "id": "S045",
      "uses": [
        "G001",
        "S044"
      ],
      "before": "Euler unit representative3 raw image normalized remainder := sub(42, 34)",
      "after": "Euler unit representative3 raw image normalized remainder = 8",
      "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 unit representative3 raw image normalized remainder，不以小數近似替代。",
          "en": "Recompute Euler unit representative3 raw image normalized remainder with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "42",
          "34"
        ],
        "output": "8",
        "label": "Euler unit representative3 raw image normalized remainder"
      }
    },
    {
      "id": "S046",
      "uses": [
        "G001",
        "S045"
      ],
      "before": "Euler unit representative3 raw image:ORIGINAL signed integer=42;modulus=17;exact ratio=42/17;floor quotient=2",
      "after": "value=q*n+remainder;normalized residue=8 in[0,17);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": "Euler ORIGINAL a times unit representative4 := mul(14, 4)",
      "after": "Euler ORIGINAL a times unit representative4 = 56",
      "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 a times unit representative4，不以小數近似替代。",
          "en": "Recompute Euler ORIGINAL a times unit representative4 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "14",
          "4"
        ],
        "output": "56",
        "label": "Euler ORIGINAL a times unit representative4"
      }
    },
    {
      "id": "S048",
      "uses": [
        "G001",
        "S047"
      ],
      "before": "Euler unit representative4 raw image exact quotient := div(56, 17)",
      "after": "Euler unit representative4 raw image exact quotient = 56/17",
      "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 unit representative4 raw image exact quotient，不以小數近似替代。",
          "en": "Recompute Euler unit representative4 raw image exact quotient with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "56",
          "17"
        ],
        "output": "56/17",
        "label": "Euler unit representative4 raw image exact quotient"
      }
    },
    {
      "id": "S049",
      "uses": [
        "G001",
        "S048"
      ],
      "before": "Euler unit representative4 raw image floor quotient multiple := mul(3, 17)",
      "after": "Euler unit representative4 raw image floor quotient multiple = 51",
      "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 unit representative4 raw image floor quotient multiple，不以小數近似替代。",
          "en": "Recompute Euler unit representative4 raw image floor quotient multiple with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "3",
          "17"
        ],
        "output": "51",
        "label": "Euler unit representative4 raw image floor quotient multiple"
      }
    },
    {
      "id": "S050",
      "uses": [
        "G001",
        "S049"
      ],
      "before": "Euler unit representative4 raw image normalized remainder := sub(56, 51)",
      "after": "Euler unit representative4 raw image normalized remainder = 5",
      "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 unit representative4 raw image normalized remainder，不以小數近似替代。",
          "en": "Recompute Euler unit representative4 raw image normalized remainder with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "56",
          "51"
        ],
        "output": "5",
        "label": "Euler unit representative4 raw image normalized remainder"
      }
    },
    {
      "id": "S051",
      "uses": [
        "G001",
        "S050"
      ],
      "before": "Euler unit representative4 raw image:ORIGINAL signed integer=56;modulus=17;exact ratio=56/17;floor quotient=3",
      "after": "value=q*n+remainder;normalized residue=5 in[0,17);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": "S052",
      "uses": [
        "G001",
        "S051"
      ],
      "before": "Euler ORIGINAL a times unit representative5 := mul(14, 5)",
      "after": "Euler ORIGINAL a times unit representative5 = 70",
      "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 a times unit representative5，不以小數近似替代。",
          "en": "Recompute Euler ORIGINAL a times unit representative5 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "14",
          "5"
        ],
        "output": "70",
        "label": "Euler ORIGINAL a times unit representative5"
      }
    },
    {
      "id": "S053",
      "uses": [
        "G001",
        "S052"
      ],
      "before": "Euler unit representative5 raw image exact quotient := div(70, 17)",
      "after": "Euler unit representative5 raw image exact quotient = 70/17",
      "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 unit representative5 raw image exact quotient，不以小數近似替代。",
          "en": "Recompute Euler unit representative5 raw image exact quotient with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "70",
          "17"
        ],
        "output": "70/17",
        "label": "Euler unit representative5 raw image exact quotient"
      }
    },
    {
      "id": "S054",
      "uses": [
        "G001",
        "S053"
      ],
      "before": "Euler unit representative5 raw image floor quotient multiple := mul(4, 17)",
      "after": "Euler unit representative5 raw image floor quotient multiple = 68",
      "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 unit representative5 raw image floor quotient multiple，不以小數近似替代。",
          "en": "Recompute Euler unit representative5 raw image floor quotient multiple with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "4",
          "17"
        ],
        "output": "68",
        "label": "Euler unit representative5 raw image floor quotient multiple"
      }
    },
    {
      "id": "S055",
      "uses": [
        "G001",
        "S054"
      ],
      "before": "Euler unit representative5 raw image normalized remainder := sub(70, 68)",
      "after": "Euler unit representative5 raw image normalized remainder = 2",
      "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 unit representative5 raw image normalized remainder，不以小數近似替代。",
          "en": "Recompute Euler unit representative5 raw image normalized remainder with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "70",
          "68"
        ],
        "output": "2",
        "label": "Euler unit representative5 raw image normalized remainder"
      }
    },
    {
      "id": "S056",
      "uses": [
        "G001",
        "S055"
      ],
      "before": "Euler unit representative5 raw image:ORIGINAL signed integer=70;modulus=17;exact ratio=70/17;floor quotient=4",
      "after": "value=q*n+remainder;normalized residue=2 in[0,17);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": "S057",
      "uses": [
        "G001",
        "S056"
      ],
      "before": "Euler ORIGINAL a times unit representative6 := mul(14, 6)",
      "after": "Euler ORIGINAL a times unit representative6 = 84",
      "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 a times unit representative6，不以小數近似替代。",
          "en": "Recompute Euler ORIGINAL a times unit representative6 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "14",
          "6"
        ],
        "output": "84",
        "label": "Euler ORIGINAL a times unit representative6"
      }
    },
    {
      "id": "S058",
      "uses": [
        "G001",
        "S057"
      ],
      "before": "Euler unit representative6 raw image exact quotient := div(84, 17)",
      "after": "Euler unit representative6 raw image exact quotient = 84/17",
      "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 unit representative6 raw image exact quotient，不以小數近似替代。",
          "en": "Recompute Euler unit representative6 raw image exact quotient with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "84",
          "17"
        ],
        "output": "84/17",
        "label": "Euler unit representative6 raw image exact quotient"
      }
    },
    {
      "id": "S059",
      "uses": [
        "G001",
        "S058"
      ],
      "before": "Euler unit representative6 raw image floor quotient multiple := mul(4, 17)",
      "after": "Euler unit representative6 raw image floor quotient multiple = 68",
      "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 unit representative6 raw image floor quotient multiple，不以小數近似替代。",
          "en": "Recompute Euler unit representative6 raw image floor quotient multiple with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "4",
          "17"
        ],
        "output": "68",
        "label": "Euler unit representative6 raw image floor quotient multiple"
      }
    },
    {
      "id": "S060",
      "uses": [
        "G001",
        "S059"
      ],
      "before": "Euler unit representative6 raw image normalized remainder := sub(84, 68)",
      "after": "Euler unit representative6 raw image normalized remainder = 16",
      "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 unit representative6 raw image normalized remainder，不以小數近似替代。",
          "en": "Recompute Euler unit representative6 raw image normalized remainder with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "84",
          "68"
        ],
        "output": "16",
        "label": "Euler unit representative6 raw image normalized remainder"
      }
    },
    {
      "id": "S061",
      "uses": [
        "G001",
        "S060"
      ],
      "before": "Euler unit representative6 raw image:ORIGINAL signed integer=84;modulus=17;exact ratio=84/17;floor quotient=4",
      "after": "value=q*n+remainder;normalized residue=16 in[0,17);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": "S062",
      "uses": [
        "G001",
        "S061"
      ],
      "before": "Euler ORIGINAL a times unit representative7 := mul(14, 7)",
      "after": "Euler ORIGINAL a times unit representative7 = 98",
      "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 a times unit representative7，不以小數近似替代。",
          "en": "Recompute Euler ORIGINAL a times unit representative7 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "14",
          "7"
        ],
        "output": "98",
        "label": "Euler ORIGINAL a times unit representative7"
      }
    },
    {
      "id": "S063",
      "uses": [
        "G001",
        "S062"
      ],
      "before": "Euler unit representative7 raw image exact quotient := div(98, 17)",
      "after": "Euler unit representative7 raw image exact quotient = 98/17",
      "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 unit representative7 raw image exact quotient，不以小數近似替代。",
          "en": "Recompute Euler unit representative7 raw image exact quotient with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "98",
          "17"
        ],
        "output": "98/17",
        "label": "Euler unit representative7 raw image exact quotient"
      }
    },
    {
      "id": "S064",
      "uses": [
        "G001",
        "S063"
      ],
      "before": "Euler unit representative7 raw image floor quotient multiple := mul(5, 17)",
      "after": "Euler unit representative7 raw image floor quotient multiple = 85",
      "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 unit representative7 raw image floor quotient multiple，不以小數近似替代。",
          "en": "Recompute Euler unit representative7 raw image floor quotient multiple with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "5",
          "17"
        ],
        "output": "85",
        "label": "Euler unit representative7 raw image floor quotient multiple"
      }
    },
    {
      "id": "S065",
      "uses": [
        "G001",
        "S064"
      ],
      "before": "Euler unit representative7 raw image normalized remainder := sub(98, 85)",
      "after": "Euler unit representative7 raw image normalized remainder = 13",
      "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 unit representative7 raw image normalized remainder，不以小數近似替代。",
          "en": "Recompute Euler unit representative7 raw image normalized remainder with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "98",
          "85"
        ],
        "output": "13",
        "label": "Euler unit representative7 raw image normalized remainder"
      }
    },
    {
      "id": "S066",
      "uses": [
        "G001",
        "S065"
      ],
      "before": "Euler unit representative7 raw image:ORIGINAL signed integer=98;modulus=17;exact ratio=98/17;floor quotient=5",
      "after": "value=q*n+remainder;normalized residue=13 in[0,17);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": "S067",
      "uses": [
        "G001",
        "S066"
      ],
      "before": "Euler ORIGINAL a times unit representative8 := mul(14, 8)",
      "after": "Euler ORIGINAL a times unit representative8 = 112",
      "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 a times unit representative8，不以小數近似替代。",
          "en": "Recompute Euler ORIGINAL a times unit representative8 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "14",
          "8"
        ],
        "output": "112",
        "label": "Euler ORIGINAL a times unit representative8"
      }
    },
    {
      "id": "S068",
      "uses": [
        "G001",
        "S067"
      ],
      "before": "Euler unit representative8 raw image exact quotient := div(112, 17)",
      "after": "Euler unit representative8 raw image exact quotient = 112/17",
      "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 unit representative8 raw image exact quotient，不以小數近似替代。",
          "en": "Recompute Euler unit representative8 raw image exact quotient with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "112",
          "17"
        ],
        "output": "112/17",
        "label": "Euler unit representative8 raw image exact quotient"
      }
    },
    {
      "id": "S069",
      "uses": [
        "G001",
        "S068"
      ],
      "before": "Euler unit representative8 raw image floor quotient multiple := mul(6, 17)",
      "after": "Euler unit representative8 raw image floor quotient multiple = 102",
      "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 unit representative8 raw image floor quotient multiple，不以小數近似替代。",
          "en": "Recompute Euler unit representative8 raw image floor quotient multiple with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "6",
          "17"
        ],
        "output": "102",
        "label": "Euler unit representative8 raw image floor quotient multiple"
      }
    },
    {
      "id": "S070",
      "uses": [
        "G001",
        "S069"
      ],
      "before": "Euler unit representative8 raw image normalized remainder := sub(112, 102)",
      "after": "Euler unit representative8 raw image normalized remainder = 10",
      "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 unit representative8 raw image normalized remainder，不以小數近似替代。",
          "en": "Recompute Euler unit representative8 raw image normalized remainder with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "112",
          "102"
        ],
        "output": "10",
        "label": "Euler unit representative8 raw image normalized remainder"
      }
    },
    {
      "id": "S071",
      "uses": [
        "G001",
        "S070"
      ],
      "before": "Euler unit representative8 raw image:ORIGINAL signed integer=112;modulus=17;exact ratio=112/17;floor quotient=6",
      "after": "value=q*n+remainder;normalized residue=10 in[0,17);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": "S072",
      "uses": [
        "G001",
        "S071"
      ],
      "before": "Euler ORIGINAL a times unit representative9 := mul(14, 9)",
      "after": "Euler ORIGINAL a times unit representative9 = 126",
      "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 a times unit representative9，不以小數近似替代。",
          "en": "Recompute Euler ORIGINAL a times unit representative9 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "14",
          "9"
        ],
        "output": "126",
        "label": "Euler ORIGINAL a times unit representative9"
      }
    },
    {
      "id": "S073",
      "uses": [
        "G001",
        "S072"
      ],
      "before": "Euler unit representative9 raw image exact quotient := div(126, 17)",
      "after": "Euler unit representative9 raw image exact quotient = 126/17",
      "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 unit representative9 raw image exact quotient，不以小數近似替代。",
          "en": "Recompute Euler unit representative9 raw image exact quotient with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "126",
          "17"
        ],
        "output": "126/17",
        "label": "Euler unit representative9 raw image exact quotient"
      }
    },
    {
      "id": "S074",
      "uses": [
        "G001",
        "S073"
      ],
      "before": "Euler unit representative9 raw image floor quotient multiple := mul(7, 17)",
      "after": "Euler unit representative9 raw image floor quotient multiple = 119",
      "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 unit representative9 raw image floor quotient multiple，不以小數近似替代。",
          "en": "Recompute Euler unit representative9 raw image floor quotient multiple with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "7",
          "17"
        ],
        "output": "119",
        "label": "Euler unit representative9 raw image floor quotient multiple"
      }
    },
    {
      "id": "S075",
      "uses": [
        "G001",
        "S074"
      ],
      "before": "Euler unit representative9 raw image normalized remainder := sub(126, 119)",
      "after": "Euler unit representative9 raw image normalized remainder = 7",
      "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 unit representative9 raw image normalized remainder，不以小數近似替代。",
          "en": "Recompute Euler unit representative9 raw image normalized remainder with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "126",
          "119"
        ],
        "output": "7",
        "label": "Euler unit representative9 raw image normalized remainder"
      }
    },
    {
      "id": "S076",
      "uses": [
        "G001",
        "S075"
      ],
      "before": "Euler unit representative9 raw image:ORIGINAL signed integer=126;modulus=17;exact ratio=126/17;floor quotient=7",
      "after": "value=q*n+remainder;normalized residue=7 in[0,17);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": "S077",
      "uses": [
        "G001",
        "S076"
      ],
      "before": "Euler ORIGINAL a times unit representative10 := mul(14, 10)",
      "after": "Euler ORIGINAL a times unit representative10 = 140",
      "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 a times unit representative10，不以小數近似替代。",
          "en": "Recompute Euler ORIGINAL a times unit representative10 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "14",
          "10"
        ],
        "output": "140",
        "label": "Euler ORIGINAL a times unit representative10"
      }
    },
    {
      "id": "S078",
      "uses": [
        "G001",
        "S077"
      ],
      "before": "Euler unit representative10 raw image exact quotient := div(140, 17)",
      "after": "Euler unit representative10 raw image exact quotient = 140/17",
      "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 unit representative10 raw image exact quotient，不以小數近似替代。",
          "en": "Recompute Euler unit representative10 raw image exact quotient with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "140",
          "17"
        ],
        "output": "140/17",
        "label": "Euler unit representative10 raw image exact quotient"
      }
    },
    {
      "id": "S079",
      "uses": [
        "G001",
        "S078"
      ],
      "before": "Euler unit representative10 raw image floor quotient multiple := mul(8, 17)",
      "after": "Euler unit representative10 raw image floor quotient multiple = 136",
      "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 unit representative10 raw image floor quotient multiple，不以小數近似替代。",
          "en": "Recompute Euler unit representative10 raw image floor quotient multiple with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "8",
          "17"
        ],
        "output": "136",
        "label": "Euler unit representative10 raw image floor quotient multiple"
      }
    },
    {
      "id": "S080",
      "uses": [
        "G001",
        "S079"
      ],
      "before": "Euler unit representative10 raw image normalized remainder := sub(140, 136)",
      "after": "Euler unit representative10 raw image normalized remainder = 4",
      "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 unit representative10 raw image normalized remainder，不以小數近似替代。",
          "en": "Recompute Euler unit representative10 raw image normalized remainder with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "140",
          "136"
        ],
        "output": "4",
        "label": "Euler unit representative10 raw image normalized remainder"
      }
    },
    {
      "id": "S081",
      "uses": [
        "G001",
        "S080"
      ],
      "before": "Euler unit representative10 raw image:ORIGINAL signed integer=140;modulus=17;exact ratio=140/17;floor quotient=8",
      "after": "value=q*n+remainder;normalized residue=4 in[0,17);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": "S082",
      "uses": [
        "G001",
        "S081"
      ],
      "before": "Euler ORIGINAL a times unit representative11 := mul(14, 11)",
      "after": "Euler ORIGINAL a times unit representative11 = 154",
      "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 a times unit representative11，不以小數近似替代。",
          "en": "Recompute Euler ORIGINAL a times unit representative11 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "14",
          "11"
        ],
        "output": "154",
        "label": "Euler ORIGINAL a times unit representative11"
      }
    },
    {
      "id": "S083",
      "uses": [
        "G001",
        "S082"
      ],
      "before": "Euler unit representative11 raw image exact quotient := div(154, 17)",
      "after": "Euler unit representative11 raw image exact quotient = 154/17",
      "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 unit representative11 raw image exact quotient，不以小數近似替代。",
          "en": "Recompute Euler unit representative11 raw image exact quotient with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "154",
          "17"
        ],
        "output": "154/17",
        "label": "Euler unit representative11 raw image exact quotient"
      }
    },
    {
      "id": "S084",
      "uses": [
        "G001",
        "S083"
      ],
      "before": "Euler unit representative11 raw image floor quotient multiple := mul(9, 17)",
      "after": "Euler unit representative11 raw image floor quotient multiple = 153",
      "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 unit representative11 raw image floor quotient multiple，不以小數近似替代。",
          "en": "Recompute Euler unit representative11 raw image floor quotient multiple with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "9",
          "17"
        ],
        "output": "153",
        "label": "Euler unit representative11 raw image floor quotient multiple"
      }
    },
    {
      "id": "S085",
      "uses": [
        "G001",
        "S084"
      ],
      "before": "Euler unit representative11 raw image normalized remainder := sub(154, 153)",
      "after": "Euler unit representative11 raw image 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 unit representative11 raw image normalized remainder，不以小數近似替代。",
          "en": "Recompute Euler unit representative11 raw image normalized remainder with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "154",
          "153"
        ],
        "output": "1",
        "label": "Euler unit representative11 raw image normalized remainder"
      }
    },
    {
      "id": "S086",
      "uses": [
        "G001",
        "S085"
      ],
      "before": "Euler unit representative11 raw image:ORIGINAL signed integer=154;modulus=17;exact ratio=154/17;floor quotient=9",
      "after": "value=q*n+remainder;normalized residue=1 in[0,17);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": "S087",
      "uses": [
        "G001",
        "S086"
      ],
      "before": "Euler ORIGINAL a times unit representative12 := mul(14, 12)",
      "after": "Euler ORIGINAL a times unit representative12 = 168",
      "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 a times unit representative12，不以小數近似替代。",
          "en": "Recompute Euler ORIGINAL a times unit representative12 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "14",
          "12"
        ],
        "output": "168",
        "label": "Euler ORIGINAL a times unit representative12"
      }
    },
    {
      "id": "S088",
      "uses": [
        "G001",
        "S087"
      ],
      "before": "Euler unit representative12 raw image exact quotient := div(168, 17)",
      "after": "Euler unit representative12 raw image exact quotient = 168/17",
      "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 unit representative12 raw image exact quotient，不以小數近似替代。",
          "en": "Recompute Euler unit representative12 raw image exact quotient with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "168",
          "17"
        ],
        "output": "168/17",
        "label": "Euler unit representative12 raw image exact quotient"
      }
    },
    {
      "id": "S089",
      "uses": [
        "G001",
        "S088"
      ],
      "before": "Euler unit representative12 raw image floor quotient multiple := mul(9, 17)",
      "after": "Euler unit representative12 raw image floor quotient multiple = 153",
      "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 unit representative12 raw image floor quotient multiple，不以小數近似替代。",
          "en": "Recompute Euler unit representative12 raw image floor quotient multiple with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "9",
          "17"
        ],
        "output": "153",
        "label": "Euler unit representative12 raw image floor quotient multiple"
      }
    },
    {
      "id": "S090",
      "uses": [
        "G001",
        "S089"
      ],
      "before": "Euler unit representative12 raw image normalized remainder := sub(168, 153)",
      "after": "Euler unit representative12 raw image normalized remainder = 15",
      "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 unit representative12 raw image normalized remainder，不以小數近似替代。",
          "en": "Recompute Euler unit representative12 raw image normalized remainder with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "168",
          "153"
        ],
        "output": "15",
        "label": "Euler unit representative12 raw image normalized remainder"
      }
    },
    {
      "id": "S091",
      "uses": [
        "G001",
        "S090"
      ],
      "before": "Euler unit representative12 raw image:ORIGINAL signed integer=168;modulus=17;exact ratio=168/17;floor quotient=9",
      "after": "value=q*n+remainder;normalized residue=15 in[0,17);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": "S092",
      "uses": [
        "G001",
        "S091"
      ],
      "before": "Euler ORIGINAL a times unit representative13 := mul(14, 13)",
      "after": "Euler ORIGINAL a times unit representative13 = 182",
      "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 a times unit representative13，不以小數近似替代。",
          "en": "Recompute Euler ORIGINAL a times unit representative13 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "14",
          "13"
        ],
        "output": "182",
        "label": "Euler ORIGINAL a times unit representative13"
      }
    },
    {
      "id": "S093",
      "uses": [
        "G001",
        "S092"
      ],
      "before": "Euler unit representative13 raw image exact quotient := div(182, 17)",
      "after": "Euler unit representative13 raw image exact quotient = 182/17",
      "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 unit representative13 raw image exact quotient，不以小數近似替代。",
          "en": "Recompute Euler unit representative13 raw image exact quotient with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "182",
          "17"
        ],
        "output": "182/17",
        "label": "Euler unit representative13 raw image exact quotient"
      }
    },
    {
      "id": "S094",
      "uses": [
        "G001",
        "S093"
      ],
      "before": "Euler unit representative13 raw image floor quotient multiple := mul(10, 17)",
      "after": "Euler unit representative13 raw image floor quotient multiple = 170",
      "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 unit representative13 raw image floor quotient multiple，不以小數近似替代。",
          "en": "Recompute Euler unit representative13 raw image floor quotient multiple with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "10",
          "17"
        ],
        "output": "170",
        "label": "Euler unit representative13 raw image floor quotient multiple"
      }
    },
    {
      "id": "S095",
      "uses": [
        "G001",
        "S094"
      ],
      "before": "Euler unit representative13 raw image normalized remainder := sub(182, 170)",
      "after": "Euler unit representative13 raw image normalized remainder = 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 unit representative13 raw image normalized remainder，不以小數近似替代。",
          "en": "Recompute Euler unit representative13 raw image normalized remainder with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "182",
          "170"
        ],
        "output": "12",
        "label": "Euler unit representative13 raw image normalized remainder"
      }
    },
    {
      "id": "S096",
      "uses": [
        "G001",
        "S095"
      ],
      "before": "Euler unit representative13 raw image:ORIGINAL signed integer=182;modulus=17;exact ratio=182/17;floor quotient=10",
      "after": "value=q*n+remainder;normalized residue=12 in[0,17);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": "S097",
      "uses": [
        "G001",
        "S096"
      ],
      "before": "Euler ORIGINAL a times unit representative14 := mul(14, 14)",
      "after": "Euler ORIGINAL a times unit representative14 = 196",
      "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 a times unit representative14，不以小數近似替代。",
          "en": "Recompute Euler ORIGINAL a times unit representative14 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "14",
          "14"
        ],
        "output": "196",
        "label": "Euler ORIGINAL a times unit representative14"
      }
    },
    {
      "id": "S098",
      "uses": [
        "G001",
        "S097"
      ],
      "before": "Euler unit representative14 raw image exact quotient := div(196, 17)",
      "after": "Euler unit representative14 raw image exact quotient = 196/17",
      "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 unit representative14 raw image exact quotient，不以小數近似替代。",
          "en": "Recompute Euler unit representative14 raw image exact quotient with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "196",
          "17"
        ],
        "output": "196/17",
        "label": "Euler unit representative14 raw image exact quotient"
      }
    },
    {
      "id": "S099",
      "uses": [
        "G001",
        "S098"
      ],
      "before": "Euler unit representative14 raw image floor quotient multiple := mul(11, 17)",
      "after": "Euler unit representative14 raw image floor quotient multiple = 187",
      "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 unit representative14 raw image floor quotient multiple，不以小數近似替代。",
          "en": "Recompute Euler unit representative14 raw image floor quotient multiple with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "11",
          "17"
        ],
        "output": "187",
        "label": "Euler unit representative14 raw image floor quotient multiple"
      }
    },
    {
      "id": "S100",
      "uses": [
        "G001",
        "S099"
      ],
      "before": "Euler unit representative14 raw image normalized remainder := sub(196, 187)",
      "after": "Euler unit representative14 raw image 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 unit representative14 raw image normalized remainder，不以小數近似替代。",
          "en": "Recompute Euler unit representative14 raw image normalized remainder with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "196",
          "187"
        ],
        "output": "9",
        "label": "Euler unit representative14 raw image normalized remainder"
      }
    },
    {
      "id": "S101",
      "uses": [
        "G001",
        "S100"
      ],
      "before": "Euler unit representative14 raw image:ORIGINAL signed integer=196;modulus=17;exact ratio=196/17;floor quotient=11",
      "after": "value=q*n+remainder;normalized residue=9 in[0,17);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": "S102",
      "uses": [
        "G001",
        "S101"
      ],
      "before": "Euler ORIGINAL a times unit representative15 := mul(14, 15)",
      "after": "Euler ORIGINAL a times unit representative15 = 210",
      "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 a times unit representative15，不以小數近似替代。",
          "en": "Recompute Euler ORIGINAL a times unit representative15 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "14",
          "15"
        ],
        "output": "210",
        "label": "Euler ORIGINAL a times unit representative15"
      }
    },
    {
      "id": "S103",
      "uses": [
        "G001",
        "S102"
      ],
      "before": "Euler unit representative15 raw image exact quotient := div(210, 17)",
      "after": "Euler unit representative15 raw image exact quotient = 210/17",
      "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 unit representative15 raw image exact quotient，不以小數近似替代。",
          "en": "Recompute Euler unit representative15 raw image exact quotient with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "210",
          "17"
        ],
        "output": "210/17",
        "label": "Euler unit representative15 raw image exact quotient"
      }
    },
    {
      "id": "S104",
      "uses": [
        "G001",
        "S103"
      ],
      "before": "Euler unit representative15 raw image floor quotient multiple := mul(12, 17)",
      "after": "Euler unit representative15 raw image floor quotient multiple = 204",
      "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 unit representative15 raw image floor quotient multiple，不以小數近似替代。",
          "en": "Recompute Euler unit representative15 raw image floor quotient multiple with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "12",
          "17"
        ],
        "output": "204",
        "label": "Euler unit representative15 raw image floor quotient multiple"
      }
    },
    {
      "id": "S105",
      "uses": [
        "G001",
        "S104"
      ],
      "before": "Euler unit representative15 raw image normalized remainder := sub(210, 204)",
      "after": "Euler unit representative15 raw image normalized remainder = 6",
      "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 unit representative15 raw image normalized remainder，不以小數近似替代。",
          "en": "Recompute Euler unit representative15 raw image normalized remainder with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "210",
          "204"
        ],
        "output": "6",
        "label": "Euler unit representative15 raw image normalized remainder"
      }
    },
    {
      "id": "S106",
      "uses": [
        "G001",
        "S105"
      ],
      "before": "Euler unit representative15 raw image:ORIGINAL signed integer=210;modulus=17;exact ratio=210/17;floor quotient=12",
      "after": "value=q*n+remainder;normalized residue=6 in[0,17);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": "S107",
      "uses": [
        "G001",
        "S106"
      ],
      "before": "Euler ORIGINAL a times unit representative16 := mul(14, 16)",
      "after": "Euler ORIGINAL a times unit representative16 = 224",
      "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 a times unit representative16，不以小數近似替代。",
          "en": "Recompute Euler ORIGINAL a times unit representative16 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "14",
          "16"
        ],
        "output": "224",
        "label": "Euler ORIGINAL a times unit representative16"
      }
    },
    {
      "id": "S108",
      "uses": [
        "G001",
        "S107"
      ],
      "before": "Euler unit representative16 raw image exact quotient := div(224, 17)",
      "after": "Euler unit representative16 raw image exact quotient = 224/17",
      "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 unit representative16 raw image exact quotient，不以小數近似替代。",
          "en": "Recompute Euler unit representative16 raw image exact quotient with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "224",
          "17"
        ],
        "output": "224/17",
        "label": "Euler unit representative16 raw image exact quotient"
      }
    },
    {
      "id": "S109",
      "uses": [
        "G001",
        "S108"
      ],
      "before": "Euler unit representative16 raw image floor quotient multiple := mul(13, 17)",
      "after": "Euler unit representative16 raw image floor quotient multiple = 221",
      "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 unit representative16 raw image floor quotient multiple，不以小數近似替代。",
          "en": "Recompute Euler unit representative16 raw image floor quotient multiple with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "13",
          "17"
        ],
        "output": "221",
        "label": "Euler unit representative16 raw image floor quotient multiple"
      }
    },
    {
      "id": "S110",
      "uses": [
        "G001",
        "S109"
      ],
      "before": "Euler unit representative16 raw image normalized remainder := sub(224, 221)",
      "after": "Euler unit representative16 raw image 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 unit representative16 raw image normalized remainder，不以小數近似替代。",
          "en": "Recompute Euler unit representative16 raw image normalized remainder with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "224",
          "221"
        ],
        "output": "3",
        "label": "Euler unit representative16 raw image normalized remainder"
      }
    },
    {
      "id": "S111",
      "uses": [
        "G001",
        "S110"
      ],
      "before": "Euler unit representative16 raw image:ORIGINAL signed integer=224;modulus=17;exact ratio=224/17;floor quotient=13",
      "after": "value=q*n+remainder;normalized residue=3 in[0,17);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": "S112",
      "uses": [
        "G001",
        "S111"
      ],
      "before": "COMPLETE original increasing U(n)=[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16];actual images in ORIGINAL unit order=[14, 11, 8, 5, 2, 16, 13, 10, 7, 4, 1, 15, 12, 9, 6, 3]",
      "after": "SOURCE output sorts the mapped list numerically=[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16];full-list equality=是;not a claim each point is fixed",
      "relation": "implies",
      "rule_id": "euler_sorted_mapped_units_not_original_image_order_or_identity",
      "conditions": [],
      "explanation": {
        "zh-Hant": "來源第二欄是排序後的完整像清單，非原unit順序的逐點像；相等證permutation，不代表乘a是identity。原始點像在本步另列，不靜默混同。",
        "en": "The source's second field is the sorted complete image list,not images in original unit order.Equal lists certify a permutation,not that multiplication by a fixes every point.The raw images are listed separately here."
      },
      "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": [
      [
        1,
        2,
        3,
        4,
        5,
        6,
        7,
        8,
        9,
        10,
        11,
        12,
        13,
        14,
        15,
        16
      ],
      [
        1,
        2,
        3,
        4,
        5,
        6,
        7,
        8,
        9,
        10,
        11,
        12,
        13,
        14,
        15,
        16
      ],
      true,
      "multiplication_by_a_permutes_reduced_residues"
    ]
  },
  "conclusion": {
    "canonical_value": "([1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], 是, multiplication_by_a_permutes_reduced_residues)",
    "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": 38,
      "original_line_sha256": "087ed0551e78742047dd1c7374e134cb8af84f23405c8e069fb0c56b3a51d9da",
      "question_compact_sha256": "40bffe11fd1e3ac7fd81121eb15a853e529a769d13802c82dabd12e5bb994fee",
      "english_row_compact_sha256": "851e9f239daa1c7be98fc3930e0c8613ddbd26918d8a1bf9e7b6838e8d8cd75e",
      "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": "6bdc3c8253e8d934253bceae6b4f83f22095d7ae7fcdf96f78c16d1e5f0dbbee",
    "original_candidate_line_sha256": "6bdc3c8253e8d934253bceae6b4f83f22095d7ae7fcdf96f78c16d1e5f0dbbee",
    "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": 112,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "d2e41b5b22a82579b527555f0b288db8437deb1f3d36af047266c057dc32ca7c",
      "arithmetic_steps": 72
    },
    "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."
  }
}