{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261014-advance-20261101-r06-euler-totient-function-0080",
  "pilot_number": "R0032-16655",
  "number": 16655,
  "family_id": "qf-mod32-totient-multiplicativity-instance",
  "title": {
    "zh-Hant": "第 16655 題：Euler φ函數",
    "en": "Question 16655: Totient Multiplicativity Requires the Coprimality Premise"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261014-advance-20261101-r06-euler-totient-function-0080",
    "batch_id": "batch-20261014-advance-20261101-r06-euler-totient-function-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-euler-totient-function"
    ],
    "title_zh": "Euler φ函數",
    "stem_zh": "正整數與完整1到n代表約定。操作=totient-multiplicativity-instance；資料={\"m\":2,\"n\":11}。請依 [m,n,phi_m,phi_n,phi_mn,product,equal,boundary] 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(2, 11, 1, 10, 10, 10, 是, instance_only_with_coprimality)",
      "explanation_zh": "Wave5根節點核對=(2, 11, 1, 10, 10, 10, 是, instance_only_with_coprimality)；依完整明示資料重算；精確計數gcd等於1的代表並保留phi(1)=1；乘法性只在互質前提下核指定實例。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 1,
      "skills": [
        "Euler φ函數"
      ]
    },
    "variant": {
      "family_id": "qf-mod32-totient-multiplicativity-instance",
      "seed": 202611016080,
      "parameters": {
        "m": 2,
        "n": 11,
        "domain": "Exact_positive_integer_totient",
        "operation": "totient-multiplicativity-instance",
        "semantic_template": "qf-mod32-totient-multiplicativity-instance/{\"m\":2,\"n\":11}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_finite_or_symbolic_model",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "候選須由主端獨立重算；不構成人工審定、一般compactness／a.s.定理／maximum matching／語言等價、Lean或形式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-20261101-r06-euler-totient-function-0080",
    "batch_id": "batch-20261014-advance-20261101-r06-euler-totient-function-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "f0912a061666c95346617ae069f296c6b057c2a4a9dda97fd0740a94d168024b",
      "family_id": "qf-mod32-totient-multiplicativity-instance",
      "seed": 202611016080,
      "semantic_template": "qf-mod32-totient-multiplicativity-instance/{\"m\":2,\"n\":11}"
    },
    "fields": {
      "title": "Totient Multiplicativity Requires the Coprimality Premise",
      "stem": "Positive integers m=2, n=11 must satisfy gcd(m,n)=1. Enumerate each complete 1-through-modulus representative range. Give (m,n,phi_m,phi_n,phi_mn,product,equal?,instance_only_with_coprimality).",
      "answer": {
        "kind": "expression",
        "value": "(2, 11, 1, 10, 10, 10, Yes, instance_only_with_coprimality)",
        "explanation": "Count all three totients exactly and compare phi(m*n) with phi(m)*phi(n). Reject noncoprime inputs rather than presenting an unrestricted multiplicative law; this result checks the supplied instance only. The exact result, retaining the requested order, is (2, 11, 1, 10, 10, 10, Yes, instance_only_with_coprimality)."
      },
      "skills": [
        "Totient Multiplicativity Requires the Coprimality Premise"
      ],
      "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": "正整數與完整1到n代表約定。操作=totient-multiplicativity-instance；資料={\"m\":2,\"n\":11}。請依 [m,n,phi_m,phi_n,phi_mn,product,equal,boundary] 欄序回答。",
        "en": "Positive integers m=2, n=11 must satisfy gcd(m,n)=1. Enumerate each complete 1-through-modulus representative range. Give (m,n,phi_m,phi_n,phi_mn,product,equal?,instance_only_with_coprimality)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "totient multiplicativity coprimality premise:original nonnegative Euclidean inputs=[2, 11]",
      "after": "division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[2, 11, 0, 2], [11, 2, 5, 1], [2, 1, 2, 0]];terminal positive gcd=1",
      "relation": "implies",
      "rule_id": "totient_Euclidean_division_chain_preserves_common_divisors",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每列明列下一步a=q·b+r及餘數界，(a,b)與(b,r)有同一批公因數；反覆替換直到第二項0，最後非零數即gcd。不是浮點近似或只報答案。",
        "en": "Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer."
      },
      "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": "totient multiplicativity original modulus product := mul(2, 11)",
      "after": "totient multiplicativity original modulus product = 22",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始欄位與已核中間量精確計算，保留索引、分母、符號與運算角色。",
        "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and operation roles."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 totient multiplicativity original modulus product，不以小數近似替代。",
          "en": "Recompute totient multiplicativity original modulus product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "11"
        ],
        "output": "22",
        "label": "totient multiplicativity original modulus product"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "left modulus representative1:original nonnegative Euclidean inputs=[1, 2]",
      "after": "division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[1, 2, 0, 1], [2, 1, 2, 0]];terminal positive gcd=1",
      "relation": "implies",
      "rule_id": "totient_Euclidean_division_chain_preserves_common_divisors",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每列明列下一步a=q·b+r及餘數界，(a,b)與(b,r)有同一批公因數；反覆替換直到第二項0，最後非零數即gcd。不是浮點近似或只報答案。",
        "en": "Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer."
      },
      "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": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "left modulus:original representative k=1 in1..2;verified gcd(k,n)=1",
      "after": "coprime indicator=1;include representative=是",
      "relation": "implies",
      "rule_id": "totient_every_original_representative_unit_gcd_test",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按1到n完整且不重複地測gcd=1，只把合格者列入。n=1時代表是1且gcd(1,1)=1，不換成另一個0到n−1清單。",
        "en": "Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list."
      },
      "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": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "left modulus representative2:original nonnegative Euclidean inputs=[2, 2]",
      "after": "division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[2, 2, 1, 0]];terminal positive gcd=2",
      "relation": "implies",
      "rule_id": "totient_Euclidean_division_chain_preserves_common_divisors",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每列明列下一步a=q·b+r及餘數界，(a,b)與(b,r)有同一批公因數；反覆替換直到第二項0，最後非零數即gcd。不是浮點近似或只報答案。",
        "en": "Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer."
      },
      "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": "left modulus:original representative k=2 in1..2;verified gcd(k,n)=2",
      "after": "coprime indicator=0;include representative=否",
      "relation": "implies",
      "rule_id": "totient_every_original_representative_unit_gcd_test",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按1到n完整且不重複地測gcd=1，只把合格者列入。n=1時代表是1且gcd(1,1)=1，不換成另一個0到n−1清單。",
        "en": "Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list."
      },
      "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": "left modulus exact coprime indicator count := sum(1, 0)",
      "after": "left modulus exact coprime indicator count = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始欄位與已核中間量精確計算，保留索引、分母、符號與運算角色。",
        "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and operation roles."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 left modulus exact coprime indicator count，不以小數近似替代。",
          "en": "Recompute left modulus exact coprime indicator count with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1",
          "0"
        ],
        "output": "1",
        "label": "left modulus exact coprime indicator count"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "left modulus:complete representative range1..2;all selected coprime representatives=[1]",
      "after": "phi(2)=1;representatives_1_through_n",
      "relation": "implies",
      "rule_id": "totient_complete_finite_count_not_sample_formula",
      "conditions": [],
      "explanation": {
        "zh-Hant": "全部指示量相加得到精確個數，這是指定n的完整有限計數，不從幾個樣本外推一般公式。",
        "en": "Sum every indicator for the exact count.This is complete finite counting for the supplied n,not a general formula extrapolated from samples."
      },
      "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": "right modulus representative1:original nonnegative Euclidean inputs=[1, 11]",
      "after": "division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[1, 11, 0, 1], [11, 1, 11, 0]];terminal positive gcd=1",
      "relation": "implies",
      "rule_id": "totient_Euclidean_division_chain_preserves_common_divisors",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每列明列下一步a=q·b+r及餘數界，(a,b)與(b,r)有同一批公因數；反覆替換直到第二項0，最後非零數即gcd。不是浮點近似或只報答案。",
        "en": "Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer."
      },
      "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": "right modulus:original representative k=1 in1..11;verified gcd(k,n)=1",
      "after": "coprime indicator=1;include representative=是",
      "relation": "implies",
      "rule_id": "totient_every_original_representative_unit_gcd_test",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按1到n完整且不重複地測gcd=1，只把合格者列入。n=1時代表是1且gcd(1,1)=1，不換成另一個0到n−1清單。",
        "en": "Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list."
      },
      "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": "right modulus representative2:original nonnegative Euclidean inputs=[2, 11]",
      "after": "division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[2, 11, 0, 2], [11, 2, 5, 1], [2, 1, 2, 0]];terminal positive gcd=1",
      "relation": "implies",
      "rule_id": "totient_Euclidean_division_chain_preserves_common_divisors",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每列明列下一步a=q·b+r及餘數界，(a,b)與(b,r)有同一批公因數；反覆替換直到第二項0，最後非零數即gcd。不是浮點近似或只報答案。",
        "en": "Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer."
      },
      "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": "right modulus:original representative k=2 in1..11;verified gcd(k,n)=1",
      "after": "coprime indicator=1;include representative=是",
      "relation": "implies",
      "rule_id": "totient_every_original_representative_unit_gcd_test",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按1到n完整且不重複地測gcd=1，只把合格者列入。n=1時代表是1且gcd(1,1)=1，不換成另一個0到n−1清單。",
        "en": "Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list."
      },
      "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": "right modulus representative3:original nonnegative Euclidean inputs=[3, 11]",
      "after": "division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[3, 11, 0, 3], [11, 3, 3, 2], [3, 2, 1, 1], [2, 1, 2, 0]];terminal positive gcd=1",
      "relation": "implies",
      "rule_id": "totient_Euclidean_division_chain_preserves_common_divisors",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每列明列下一步a=q·b+r及餘數界，(a,b)與(b,r)有同一批公因數；反覆替換直到第二項0，最後非零數即gcd。不是浮點近似或只報答案。",
        "en": "Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer."
      },
      "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": "right modulus:original representative k=3 in1..11;verified gcd(k,n)=1",
      "after": "coprime indicator=1;include representative=是",
      "relation": "implies",
      "rule_id": "totient_every_original_representative_unit_gcd_test",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按1到n完整且不重複地測gcd=1，只把合格者列入。n=1時代表是1且gcd(1,1)=1，不換成另一個0到n−1清單。",
        "en": "Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list."
      },
      "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": "right modulus representative4:original nonnegative Euclidean inputs=[4, 11]",
      "after": "division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[4, 11, 0, 4], [11, 4, 2, 3], [4, 3, 1, 1], [3, 1, 3, 0]];terminal positive gcd=1",
      "relation": "implies",
      "rule_id": "totient_Euclidean_division_chain_preserves_common_divisors",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每列明列下一步a=q·b+r及餘數界，(a,b)與(b,r)有同一批公因數；反覆替換直到第二項0，最後非零數即gcd。不是浮點近似或只報答案。",
        "en": "Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer."
      },
      "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": "right modulus:original representative k=4 in1..11;verified gcd(k,n)=1",
      "after": "coprime indicator=1;include representative=是",
      "relation": "implies",
      "rule_id": "totient_every_original_representative_unit_gcd_test",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按1到n完整且不重複地測gcd=1，只把合格者列入。n=1時代表是1且gcd(1,1)=1，不換成另一個0到n−1清單。",
        "en": "Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list."
      },
      "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": "right modulus representative5:original nonnegative Euclidean inputs=[5, 11]",
      "after": "division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[5, 11, 0, 5], [11, 5, 2, 1], [5, 1, 5, 0]];terminal positive gcd=1",
      "relation": "implies",
      "rule_id": "totient_Euclidean_division_chain_preserves_common_divisors",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每列明列下一步a=q·b+r及餘數界，(a,b)與(b,r)有同一批公因數；反覆替換直到第二項0，最後非零數即gcd。不是浮點近似或只報答案。",
        "en": "Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer."
      },
      "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": "right modulus:original representative k=5 in1..11;verified gcd(k,n)=1",
      "after": "coprime indicator=1;include representative=是",
      "relation": "implies",
      "rule_id": "totient_every_original_representative_unit_gcd_test",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按1到n完整且不重複地測gcd=1，只把合格者列入。n=1時代表是1且gcd(1,1)=1，不換成另一個0到n−1清單。",
        "en": "Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list."
      },
      "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": "right modulus representative6:original nonnegative Euclidean inputs=[6, 11]",
      "after": "division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[6, 11, 0, 6], [11, 6, 1, 5], [6, 5, 1, 1], [5, 1, 5, 0]];terminal positive gcd=1",
      "relation": "implies",
      "rule_id": "totient_Euclidean_division_chain_preserves_common_divisors",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每列明列下一步a=q·b+r及餘數界，(a,b)與(b,r)有同一批公因數；反覆替換直到第二項0，最後非零數即gcd。不是浮點近似或只報答案。",
        "en": "Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer."
      },
      "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": "right modulus:original representative k=6 in1..11;verified gcd(k,n)=1",
      "after": "coprime indicator=1;include representative=是",
      "relation": "implies",
      "rule_id": "totient_every_original_representative_unit_gcd_test",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按1到n完整且不重複地測gcd=1，只把合格者列入。n=1時代表是1且gcd(1,1)=1，不換成另一個0到n−1清單。",
        "en": "Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list."
      },
      "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": "right modulus representative7:original nonnegative Euclidean inputs=[7, 11]",
      "after": "division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[7, 11, 0, 7], [11, 7, 1, 4], [7, 4, 1, 3], [4, 3, 1, 1], [3, 1, 3, 0]];terminal positive gcd=1",
      "relation": "implies",
      "rule_id": "totient_Euclidean_division_chain_preserves_common_divisors",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每列明列下一步a=q·b+r及餘數界，(a,b)與(b,r)有同一批公因數；反覆替換直到第二項0，最後非零數即gcd。不是浮點近似或只報答案。",
        "en": "Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer."
      },
      "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": "right modulus:original representative k=7 in1..11;verified gcd(k,n)=1",
      "after": "coprime indicator=1;include representative=是",
      "relation": "implies",
      "rule_id": "totient_every_original_representative_unit_gcd_test",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按1到n完整且不重複地測gcd=1，只把合格者列入。n=1時代表是1且gcd(1,1)=1，不換成另一個0到n−1清單。",
        "en": "Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list."
      },
      "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": "right modulus representative8:original nonnegative Euclidean inputs=[8, 11]",
      "after": "division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[8, 11, 0, 8], [11, 8, 1, 3], [8, 3, 2, 2], [3, 2, 1, 1], [2, 1, 2, 0]];terminal positive gcd=1",
      "relation": "implies",
      "rule_id": "totient_Euclidean_division_chain_preserves_common_divisors",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每列明列下一步a=q·b+r及餘數界，(a,b)與(b,r)有同一批公因數；反覆替換直到第二項0，最後非零數即gcd。不是浮點近似或只報答案。",
        "en": "Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer."
      },
      "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": "right modulus:original representative k=8 in1..11;verified gcd(k,n)=1",
      "after": "coprime indicator=1;include representative=是",
      "relation": "implies",
      "rule_id": "totient_every_original_representative_unit_gcd_test",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按1到n完整且不重複地測gcd=1，只把合格者列入。n=1時代表是1且gcd(1,1)=1，不換成另一個0到n−1清單。",
        "en": "Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S025",
      "uses": [
        "G001",
        "S024"
      ],
      "before": "right modulus representative9:original nonnegative Euclidean inputs=[9, 11]",
      "after": "division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[9, 11, 0, 9], [11, 9, 1, 2], [9, 2, 4, 1], [2, 1, 2, 0]];terminal positive gcd=1",
      "relation": "implies",
      "rule_id": "totient_Euclidean_division_chain_preserves_common_divisors",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每列明列下一步a=q·b+r及餘數界，(a,b)與(b,r)有同一批公因數；反覆替換直到第二項0，最後非零數即gcd。不是浮點近似或只報答案。",
        "en": "Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer."
      },
      "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": "right modulus:original representative k=9 in1..11;verified gcd(k,n)=1",
      "after": "coprime indicator=1;include representative=是",
      "relation": "implies",
      "rule_id": "totient_every_original_representative_unit_gcd_test",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按1到n完整且不重複地測gcd=1，只把合格者列入。n=1時代表是1且gcd(1,1)=1，不換成另一個0到n−1清單。",
        "en": "Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S027",
      "uses": [
        "G001",
        "S026"
      ],
      "before": "right modulus representative10:original nonnegative Euclidean inputs=[10, 11]",
      "after": "division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[10, 11, 0, 10], [11, 10, 1, 1], [10, 1, 10, 0]];terminal positive gcd=1",
      "relation": "implies",
      "rule_id": "totient_Euclidean_division_chain_preserves_common_divisors",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每列明列下一步a=q·b+r及餘數界，(a,b)與(b,r)有同一批公因數；反覆替換直到第二項0，最後非零數即gcd。不是浮點近似或只報答案。",
        "en": "Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S028",
      "uses": [
        "G001",
        "S027"
      ],
      "before": "right modulus:original representative k=10 in1..11;verified gcd(k,n)=1",
      "after": "coprime indicator=1;include representative=是",
      "relation": "implies",
      "rule_id": "totient_every_original_representative_unit_gcd_test",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按1到n完整且不重複地測gcd=1，只把合格者列入。n=1時代表是1且gcd(1,1)=1，不換成另一個0到n−1清單。",
        "en": "Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S029",
      "uses": [
        "G001",
        "S028"
      ],
      "before": "right modulus representative11:original nonnegative Euclidean inputs=[11, 11]",
      "after": "division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[11, 11, 1, 0]];terminal positive gcd=11",
      "relation": "implies",
      "rule_id": "totient_Euclidean_division_chain_preserves_common_divisors",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每列明列下一步a=q·b+r及餘數界，(a,b)與(b,r)有同一批公因數；反覆替換直到第二項0，最後非零數即gcd。不是浮點近似或只報答案。",
        "en": "Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S030",
      "uses": [
        "G001",
        "S029"
      ],
      "before": "right modulus:original representative k=11 in1..11;verified gcd(k,n)=11",
      "after": "coprime indicator=0;include representative=否",
      "relation": "implies",
      "rule_id": "totient_every_original_representative_unit_gcd_test",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按1到n完整且不重複地測gcd=1，只把合格者列入。n=1時代表是1且gcd(1,1)=1，不換成另一個0到n−1清單。",
        "en": "Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list."
      },
      "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": "right modulus exact coprime indicator count := sum(1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0)",
      "after": "right modulus exact coprime indicator count = 10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始欄位與已核中間量精確計算，保留索引、分母、符號與運算角色。",
        "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and operation roles."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 right modulus exact coprime indicator count，不以小數近似替代。",
          "en": "Recompute right modulus exact coprime indicator count with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1",
          "1",
          "1",
          "1",
          "1",
          "1",
          "1",
          "1",
          "1",
          "1",
          "0"
        ],
        "output": "10",
        "label": "right modulus exact coprime indicator count"
      }
    },
    {
      "id": "S032",
      "uses": [
        "G001",
        "S031"
      ],
      "before": "right modulus:complete representative range1..11;all selected coprime representatives=[1, 2, 3, 4, 5, 6, 7, 8, 9, 10]",
      "after": "phi(11)=10;representatives_1_through_n",
      "relation": "implies",
      "rule_id": "totient_complete_finite_count_not_sample_formula",
      "conditions": [],
      "explanation": {
        "zh-Hant": "全部指示量相加得到精確個數，這是指定n的完整有限計數，不從幾個樣本外推一般公式。",
        "en": "Sum every indicator for the exact count.This is complete finite counting for the supplied n,not a general formula extrapolated from samples."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S033",
      "uses": [
        "G001",
        "S032"
      ],
      "before": "product modulus representative1:original nonnegative Euclidean inputs=[1, 22]",
      "after": "division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[1, 22, 0, 1], [22, 1, 22, 0]];terminal positive gcd=1",
      "relation": "implies",
      "rule_id": "totient_Euclidean_division_chain_preserves_common_divisors",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每列明列下一步a=q·b+r及餘數界，(a,b)與(b,r)有同一批公因數；反覆替換直到第二項0，最後非零數即gcd。不是浮點近似或只報答案。",
        "en": "Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer."
      },
      "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": "S034",
      "uses": [
        "G001",
        "S033"
      ],
      "before": "product modulus:original representative k=1 in1..22;verified gcd(k,n)=1",
      "after": "coprime indicator=1;include representative=是",
      "relation": "implies",
      "rule_id": "totient_every_original_representative_unit_gcd_test",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按1到n完整且不重複地測gcd=1，只把合格者列入。n=1時代表是1且gcd(1,1)=1，不換成另一個0到n−1清單。",
        "en": "Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S035",
      "uses": [
        "G001",
        "S034"
      ],
      "before": "product modulus representative2:original nonnegative Euclidean inputs=[2, 22]",
      "after": "division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[2, 22, 0, 2], [22, 2, 11, 0]];terminal positive gcd=2",
      "relation": "implies",
      "rule_id": "totient_Euclidean_division_chain_preserves_common_divisors",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每列明列下一步a=q·b+r及餘數界，(a,b)與(b,r)有同一批公因數；反覆替換直到第二項0，最後非零數即gcd。不是浮點近似或只報答案。",
        "en": "Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S036",
      "uses": [
        "G001",
        "S035"
      ],
      "before": "product modulus:original representative k=2 in1..22;verified gcd(k,n)=2",
      "after": "coprime indicator=0;include representative=否",
      "relation": "implies",
      "rule_id": "totient_every_original_representative_unit_gcd_test",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按1到n完整且不重複地測gcd=1，只把合格者列入。n=1時代表是1且gcd(1,1)=1，不換成另一個0到n−1清單。",
        "en": "Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list."
      },
      "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": "product modulus representative3:original nonnegative Euclidean inputs=[3, 22]",
      "after": "division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[3, 22, 0, 3], [22, 3, 7, 1], [3, 1, 3, 0]];terminal positive gcd=1",
      "relation": "implies",
      "rule_id": "totient_Euclidean_division_chain_preserves_common_divisors",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每列明列下一步a=q·b+r及餘數界，(a,b)與(b,r)有同一批公因數；反覆替換直到第二項0，最後非零數即gcd。不是浮點近似或只報答案。",
        "en": "Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer."
      },
      "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": "S038",
      "uses": [
        "G001",
        "S037"
      ],
      "before": "product modulus:original representative k=3 in1..22;verified gcd(k,n)=1",
      "after": "coprime indicator=1;include representative=是",
      "relation": "implies",
      "rule_id": "totient_every_original_representative_unit_gcd_test",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按1到n完整且不重複地測gcd=1，只把合格者列入。n=1時代表是1且gcd(1,1)=1，不換成另一個0到n−1清單。",
        "en": "Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list."
      },
      "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": "S039",
      "uses": [
        "G001",
        "S038"
      ],
      "before": "product modulus representative4:original nonnegative Euclidean inputs=[4, 22]",
      "after": "division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[4, 22, 0, 4], [22, 4, 5, 2], [4, 2, 2, 0]];terminal positive gcd=2",
      "relation": "implies",
      "rule_id": "totient_Euclidean_division_chain_preserves_common_divisors",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每列明列下一步a=q·b+r及餘數界，(a,b)與(b,r)有同一批公因數；反覆替換直到第二項0，最後非零數即gcd。不是浮點近似或只報答案。",
        "en": "Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer."
      },
      "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": "S040",
      "uses": [
        "G001",
        "S039"
      ],
      "before": "product modulus:original representative k=4 in1..22;verified gcd(k,n)=2",
      "after": "coprime indicator=0;include representative=否",
      "relation": "implies",
      "rule_id": "totient_every_original_representative_unit_gcd_test",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按1到n完整且不重複地測gcd=1，只把合格者列入。n=1時代表是1且gcd(1,1)=1，不換成另一個0到n−1清單。",
        "en": "Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S041",
      "uses": [
        "G001",
        "S040"
      ],
      "before": "product modulus representative5:original nonnegative Euclidean inputs=[5, 22]",
      "after": "division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[5, 22, 0, 5], [22, 5, 4, 2], [5, 2, 2, 1], [2, 1, 2, 0]];terminal positive gcd=1",
      "relation": "implies",
      "rule_id": "totient_Euclidean_division_chain_preserves_common_divisors",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每列明列下一步a=q·b+r及餘數界，(a,b)與(b,r)有同一批公因數；反覆替換直到第二項0，最後非零數即gcd。不是浮點近似或只報答案。",
        "en": "Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer."
      },
      "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": "product modulus:original representative k=5 in1..22;verified gcd(k,n)=1",
      "after": "coprime indicator=1;include representative=是",
      "relation": "implies",
      "rule_id": "totient_every_original_representative_unit_gcd_test",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按1到n完整且不重複地測gcd=1，只把合格者列入。n=1時代表是1且gcd(1,1)=1，不換成另一個0到n−1清單。",
        "en": "Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list."
      },
      "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": "S043",
      "uses": [
        "G001",
        "S042"
      ],
      "before": "product modulus representative6:original nonnegative Euclidean inputs=[6, 22]",
      "after": "division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[6, 22, 0, 6], [22, 6, 3, 4], [6, 4, 1, 2], [4, 2, 2, 0]];terminal positive gcd=2",
      "relation": "implies",
      "rule_id": "totient_Euclidean_division_chain_preserves_common_divisors",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每列明列下一步a=q·b+r及餘數界，(a,b)與(b,r)有同一批公因數；反覆替換直到第二項0，最後非零數即gcd。不是浮點近似或只報答案。",
        "en": "Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer."
      },
      "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": "S044",
      "uses": [
        "G001",
        "S043"
      ],
      "before": "product modulus:original representative k=6 in1..22;verified gcd(k,n)=2",
      "after": "coprime indicator=0;include representative=否",
      "relation": "implies",
      "rule_id": "totient_every_original_representative_unit_gcd_test",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按1到n完整且不重複地測gcd=1，只把合格者列入。n=1時代表是1且gcd(1,1)=1，不換成另一個0到n−1清單。",
        "en": "Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list."
      },
      "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": "S045",
      "uses": [
        "G001",
        "S044"
      ],
      "before": "product modulus representative7:original nonnegative Euclidean inputs=[7, 22]",
      "after": "division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[7, 22, 0, 7], [22, 7, 3, 1], [7, 1, 7, 0]];terminal positive gcd=1",
      "relation": "implies",
      "rule_id": "totient_Euclidean_division_chain_preserves_common_divisors",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每列明列下一步a=q·b+r及餘數界，(a,b)與(b,r)有同一批公因數；反覆替換直到第二項0，最後非零數即gcd。不是浮點近似或只報答案。",
        "en": "Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer."
      },
      "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": "S046",
      "uses": [
        "G001",
        "S045"
      ],
      "before": "product modulus:original representative k=7 in1..22;verified gcd(k,n)=1",
      "after": "coprime indicator=1;include representative=是",
      "relation": "implies",
      "rule_id": "totient_every_original_representative_unit_gcd_test",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按1到n完整且不重複地測gcd=1，只把合格者列入。n=1時代表是1且gcd(1,1)=1，不換成另一個0到n−1清單。",
        "en": "Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list."
      },
      "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": "product modulus representative8:original nonnegative Euclidean inputs=[8, 22]",
      "after": "division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[8, 22, 0, 8], [22, 8, 2, 6], [8, 6, 1, 2], [6, 2, 3, 0]];terminal positive gcd=2",
      "relation": "implies",
      "rule_id": "totient_Euclidean_division_chain_preserves_common_divisors",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每列明列下一步a=q·b+r及餘數界，(a,b)與(b,r)有同一批公因數；反覆替換直到第二項0，最後非零數即gcd。不是浮點近似或只報答案。",
        "en": "Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S048",
      "uses": [
        "G001",
        "S047"
      ],
      "before": "product modulus:original representative k=8 in1..22;verified gcd(k,n)=2",
      "after": "coprime indicator=0;include representative=否",
      "relation": "implies",
      "rule_id": "totient_every_original_representative_unit_gcd_test",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按1到n完整且不重複地測gcd=1，只把合格者列入。n=1時代表是1且gcd(1,1)=1，不換成另一個0到n−1清單。",
        "en": "Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S049",
      "uses": [
        "G001",
        "S048"
      ],
      "before": "product modulus representative9:original nonnegative Euclidean inputs=[9, 22]",
      "after": "division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[9, 22, 0, 9], [22, 9, 2, 4], [9, 4, 2, 1], [4, 1, 4, 0]];terminal positive gcd=1",
      "relation": "implies",
      "rule_id": "totient_Euclidean_division_chain_preserves_common_divisors",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每列明列下一步a=q·b+r及餘數界，(a,b)與(b,r)有同一批公因數；反覆替換直到第二項0，最後非零數即gcd。不是浮點近似或只報答案。",
        "en": "Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S050",
      "uses": [
        "G001",
        "S049"
      ],
      "before": "product modulus:original representative k=9 in1..22;verified gcd(k,n)=1",
      "after": "coprime indicator=1;include representative=是",
      "relation": "implies",
      "rule_id": "totient_every_original_representative_unit_gcd_test",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按1到n完整且不重複地測gcd=1，只把合格者列入。n=1時代表是1且gcd(1,1)=1，不換成另一個0到n−1清單。",
        "en": "Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S051",
      "uses": [
        "G001",
        "S050"
      ],
      "before": "product modulus representative10:original nonnegative Euclidean inputs=[10, 22]",
      "after": "division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[10, 22, 0, 10], [22, 10, 2, 2], [10, 2, 5, 0]];terminal positive gcd=2",
      "relation": "implies",
      "rule_id": "totient_Euclidean_division_chain_preserves_common_divisors",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每列明列下一步a=q·b+r及餘數界，(a,b)與(b,r)有同一批公因數；反覆替換直到第二項0，最後非零數即gcd。不是浮點近似或只報答案。",
        "en": "Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer."
      },
      "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": "product modulus:original representative k=10 in1..22;verified gcd(k,n)=2",
      "after": "coprime indicator=0;include representative=否",
      "relation": "implies",
      "rule_id": "totient_every_original_representative_unit_gcd_test",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按1到n完整且不重複地測gcd=1，只把合格者列入。n=1時代表是1且gcd(1,1)=1，不換成另一個0到n−1清單。",
        "en": "Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S053",
      "uses": [
        "G001",
        "S052"
      ],
      "before": "product modulus representative11:original nonnegative Euclidean inputs=[11, 22]",
      "after": "division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[11, 22, 0, 11], [22, 11, 2, 0]];terminal positive gcd=11",
      "relation": "implies",
      "rule_id": "totient_Euclidean_division_chain_preserves_common_divisors",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每列明列下一步a=q·b+r及餘數界，(a,b)與(b,r)有同一批公因數；反覆替換直到第二項0，最後非零數即gcd。不是浮點近似或只報答案。",
        "en": "Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer."
      },
      "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": "S054",
      "uses": [
        "G001",
        "S053"
      ],
      "before": "product modulus:original representative k=11 in1..22;verified gcd(k,n)=11",
      "after": "coprime indicator=0;include representative=否",
      "relation": "implies",
      "rule_id": "totient_every_original_representative_unit_gcd_test",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按1到n完整且不重複地測gcd=1，只把合格者列入。n=1時代表是1且gcd(1,1)=1，不換成另一個0到n−1清單。",
        "en": "Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list."
      },
      "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": "S055",
      "uses": [
        "G001",
        "S054"
      ],
      "before": "product modulus representative12:original nonnegative Euclidean inputs=[12, 22]",
      "after": "division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[12, 22, 0, 12], [22, 12, 1, 10], [12, 10, 1, 2], [10, 2, 5, 0]];terminal positive gcd=2",
      "relation": "implies",
      "rule_id": "totient_Euclidean_division_chain_preserves_common_divisors",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每列明列下一步a=q·b+r及餘數界，(a,b)與(b,r)有同一批公因數；反覆替換直到第二項0，最後非零數即gcd。不是浮點近似或只報答案。",
        "en": "Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer."
      },
      "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": "S056",
      "uses": [
        "G001",
        "S055"
      ],
      "before": "product modulus:original representative k=12 in1..22;verified gcd(k,n)=2",
      "after": "coprime indicator=0;include representative=否",
      "relation": "implies",
      "rule_id": "totient_every_original_representative_unit_gcd_test",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按1到n完整且不重複地測gcd=1，只把合格者列入。n=1時代表是1且gcd(1,1)=1，不換成另一個0到n−1清單。",
        "en": "Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list."
      },
      "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": "product modulus representative13:original nonnegative Euclidean inputs=[13, 22]",
      "after": "division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[13, 22, 0, 13], [22, 13, 1, 9], [13, 9, 1, 4], [9, 4, 2, 1], [4, 1, 4, 0]];terminal positive gcd=1",
      "relation": "implies",
      "rule_id": "totient_Euclidean_division_chain_preserves_common_divisors",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每列明列下一步a=q·b+r及餘數界，(a,b)與(b,r)有同一批公因數；反覆替換直到第二項0，最後非零數即gcd。不是浮點近似或只報答案。",
        "en": "Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S058",
      "uses": [
        "G001",
        "S057"
      ],
      "before": "product modulus:original representative k=13 in1..22;verified gcd(k,n)=1",
      "after": "coprime indicator=1;include representative=是",
      "relation": "implies",
      "rule_id": "totient_every_original_representative_unit_gcd_test",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按1到n完整且不重複地測gcd=1，只把合格者列入。n=1時代表是1且gcd(1,1)=1，不換成另一個0到n−1清單。",
        "en": "Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S059",
      "uses": [
        "G001",
        "S058"
      ],
      "before": "product modulus representative14:original nonnegative Euclidean inputs=[14, 22]",
      "after": "division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[14, 22, 0, 14], [22, 14, 1, 8], [14, 8, 1, 6], [8, 6, 1, 2], [6, 2, 3, 0]];terminal positive gcd=2",
      "relation": "implies",
      "rule_id": "totient_Euclidean_division_chain_preserves_common_divisors",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每列明列下一步a=q·b+r及餘數界，(a,b)與(b,r)有同一批公因數；反覆替換直到第二項0，最後非零數即gcd。不是浮點近似或只報答案。",
        "en": "Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer."
      },
      "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": "S060",
      "uses": [
        "G001",
        "S059"
      ],
      "before": "product modulus:original representative k=14 in1..22;verified gcd(k,n)=2",
      "after": "coprime indicator=0;include representative=否",
      "relation": "implies",
      "rule_id": "totient_every_original_representative_unit_gcd_test",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按1到n完整且不重複地測gcd=1，只把合格者列入。n=1時代表是1且gcd(1,1)=1，不換成另一個0到n−1清單。",
        "en": "Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list."
      },
      "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": "S061",
      "uses": [
        "G001",
        "S060"
      ],
      "before": "product modulus representative15:original nonnegative Euclidean inputs=[15, 22]",
      "after": "division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[15, 22, 0, 15], [22, 15, 1, 7], [15, 7, 2, 1], [7, 1, 7, 0]];terminal positive gcd=1",
      "relation": "implies",
      "rule_id": "totient_Euclidean_division_chain_preserves_common_divisors",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每列明列下一步a=q·b+r及餘數界，(a,b)與(b,r)有同一批公因數；反覆替換直到第二項0，最後非零數即gcd。不是浮點近似或只報答案。",
        "en": "Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer."
      },
      "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": "product modulus:original representative k=15 in1..22;verified gcd(k,n)=1",
      "after": "coprime indicator=1;include representative=是",
      "relation": "implies",
      "rule_id": "totient_every_original_representative_unit_gcd_test",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按1到n完整且不重複地測gcd=1，只把合格者列入。n=1時代表是1且gcd(1,1)=1，不換成另一個0到n−1清單。",
        "en": "Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list."
      },
      "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": "S063",
      "uses": [
        "G001",
        "S062"
      ],
      "before": "product modulus representative16:original nonnegative Euclidean inputs=[16, 22]",
      "after": "division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[16, 22, 0, 16], [22, 16, 1, 6], [16, 6, 2, 4], [6, 4, 1, 2], [4, 2, 2, 0]];terminal positive gcd=2",
      "relation": "implies",
      "rule_id": "totient_Euclidean_division_chain_preserves_common_divisors",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每列明列下一步a=q·b+r及餘數界，(a,b)與(b,r)有同一批公因數；反覆替換直到第二項0，最後非零數即gcd。不是浮點近似或只報答案。",
        "en": "Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S064",
      "uses": [
        "G001",
        "S063"
      ],
      "before": "product modulus:original representative k=16 in1..22;verified gcd(k,n)=2",
      "after": "coprime indicator=0;include representative=否",
      "relation": "implies",
      "rule_id": "totient_every_original_representative_unit_gcd_test",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按1到n完整且不重複地測gcd=1，只把合格者列入。n=1時代表是1且gcd(1,1)=1，不換成另一個0到n−1清單。",
        "en": "Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list."
      },
      "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": "S065",
      "uses": [
        "G001",
        "S064"
      ],
      "before": "product modulus representative17:original nonnegative Euclidean inputs=[17, 22]",
      "after": "division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[17, 22, 0, 17], [22, 17, 1, 5], [17, 5, 3, 2], [5, 2, 2, 1], [2, 1, 2, 0]];terminal positive gcd=1",
      "relation": "implies",
      "rule_id": "totient_Euclidean_division_chain_preserves_common_divisors",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每列明列下一步a=q·b+r及餘數界，(a,b)與(b,r)有同一批公因數；反覆替換直到第二項0，最後非零數即gcd。不是浮點近似或只報答案。",
        "en": "Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S066",
      "uses": [
        "G001",
        "S065"
      ],
      "before": "product modulus:original representative k=17 in1..22;verified gcd(k,n)=1",
      "after": "coprime indicator=1;include representative=是",
      "relation": "implies",
      "rule_id": "totient_every_original_representative_unit_gcd_test",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按1到n完整且不重複地測gcd=1，只把合格者列入。n=1時代表是1且gcd(1,1)=1，不換成另一個0到n−1清單。",
        "en": "Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list."
      },
      "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": "product modulus representative18:original nonnegative Euclidean inputs=[18, 22]",
      "after": "division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[18, 22, 0, 18], [22, 18, 1, 4], [18, 4, 4, 2], [4, 2, 2, 0]];terminal positive gcd=2",
      "relation": "implies",
      "rule_id": "totient_Euclidean_division_chain_preserves_common_divisors",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每列明列下一步a=q·b+r及餘數界，(a,b)與(b,r)有同一批公因數；反覆替換直到第二項0，最後非零數即gcd。不是浮點近似或只報答案。",
        "en": "Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer."
      },
      "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": "S068",
      "uses": [
        "G001",
        "S067"
      ],
      "before": "product modulus:original representative k=18 in1..22;verified gcd(k,n)=2",
      "after": "coprime indicator=0;include representative=否",
      "relation": "implies",
      "rule_id": "totient_every_original_representative_unit_gcd_test",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按1到n完整且不重複地測gcd=1，只把合格者列入。n=1時代表是1且gcd(1,1)=1，不換成另一個0到n−1清單。",
        "en": "Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S069",
      "uses": [
        "G001",
        "S068"
      ],
      "before": "product modulus representative19:original nonnegative Euclidean inputs=[19, 22]",
      "after": "division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[19, 22, 0, 19], [22, 19, 1, 3], [19, 3, 6, 1], [3, 1, 3, 0]];terminal positive gcd=1",
      "relation": "implies",
      "rule_id": "totient_Euclidean_division_chain_preserves_common_divisors",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每列明列下一步a=q·b+r及餘數界，(a,b)與(b,r)有同一批公因數；反覆替換直到第二項0，最後非零數即gcd。不是浮點近似或只報答案。",
        "en": "Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S070",
      "uses": [
        "G001",
        "S069"
      ],
      "before": "product modulus:original representative k=19 in1..22;verified gcd(k,n)=1",
      "after": "coprime indicator=1;include representative=是",
      "relation": "implies",
      "rule_id": "totient_every_original_representative_unit_gcd_test",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按1到n完整且不重複地測gcd=1，只把合格者列入。n=1時代表是1且gcd(1,1)=1，不換成另一個0到n−1清單。",
        "en": "Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S071",
      "uses": [
        "G001",
        "S070"
      ],
      "before": "product modulus representative20:original nonnegative Euclidean inputs=[20, 22]",
      "after": "division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[20, 22, 0, 20], [22, 20, 1, 2], [20, 2, 10, 0]];terminal positive gcd=2",
      "relation": "implies",
      "rule_id": "totient_Euclidean_division_chain_preserves_common_divisors",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每列明列下一步a=q·b+r及餘數界，(a,b)與(b,r)有同一批公因數；反覆替換直到第二項0，最後非零數即gcd。不是浮點近似或只報答案。",
        "en": "Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer."
      },
      "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": "product modulus:original representative k=20 in1..22;verified gcd(k,n)=2",
      "after": "coprime indicator=0;include representative=否",
      "relation": "implies",
      "rule_id": "totient_every_original_representative_unit_gcd_test",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按1到n完整且不重複地測gcd=1，只把合格者列入。n=1時代表是1且gcd(1,1)=1，不換成另一個0到n−1清單。",
        "en": "Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list."
      },
      "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": "S073",
      "uses": [
        "G001",
        "S072"
      ],
      "before": "product modulus representative21:original nonnegative Euclidean inputs=[21, 22]",
      "after": "division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[21, 22, 0, 21], [22, 21, 1, 1], [21, 1, 21, 0]];terminal positive gcd=1",
      "relation": "implies",
      "rule_id": "totient_Euclidean_division_chain_preserves_common_divisors",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每列明列下一步a=q·b+r及餘數界，(a,b)與(b,r)有同一批公因數；反覆替換直到第二項0，最後非零數即gcd。不是浮點近似或只報答案。",
        "en": "Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S074",
      "uses": [
        "G001",
        "S073"
      ],
      "before": "product modulus:original representative k=21 in1..22;verified gcd(k,n)=1",
      "after": "coprime indicator=1;include representative=是",
      "relation": "implies",
      "rule_id": "totient_every_original_representative_unit_gcd_test",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按1到n完整且不重複地測gcd=1，只把合格者列入。n=1時代表是1且gcd(1,1)=1，不換成另一個0到n−1清單。",
        "en": "Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list."
      },
      "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": "S075",
      "uses": [
        "G001",
        "S074"
      ],
      "before": "product modulus representative22:original nonnegative Euclidean inputs=[22, 22]",
      "after": "division rows[a,b,q,r] with a=q*b+r and0<=r<b=[[22, 22, 1, 0]];terminal positive gcd=22",
      "relation": "implies",
      "rule_id": "totient_Euclidean_division_chain_preserves_common_divisors",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每列明列下一步a=q·b+r及餘數界，(a,b)與(b,r)有同一批公因數；反覆替換直到第二項0，最後非零數即gcd。不是浮點近似或只報答案。",
        "en": "Each row gives a=q*b+r with its remainder bound.The pairs(a,b)and(b,r)have exactly the same common divisors.Repeat until the second entry is0;the last nonzero value is the gcd.This is an exact chain,not a floating-point approximation or an unexplained answer."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S076",
      "uses": [
        "G001",
        "S075"
      ],
      "before": "product modulus:original representative k=22 in1..22;verified gcd(k,n)=22",
      "after": "coprime indicator=0;include representative=否",
      "relation": "implies",
      "rule_id": "totient_every_original_representative_unit_gcd_test",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按1到n完整且不重複地測gcd=1，只把合格者列入。n=1時代表是1且gcd(1,1)=1，不換成另一個0到n−1清單。",
        "en": "Test gcd=1 for every original representative1 through n exactly once.Include only qualifying representatives.For n=1 the representative is1 and gcd(1,1)=1;do not switch to a different0-through-n−1 list."
      },
      "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": "product modulus exact coprime indicator count := sum(1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0)",
      "after": "product modulus exact coprime indicator count = 10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始欄位與已核中間量精確計算，保留索引、分母、符號與運算角色。",
        "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and operation roles."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 product modulus exact coprime indicator count，不以小數近似替代。",
          "en": "Recompute product modulus exact coprime indicator count with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1",
          "0",
          "1",
          "0",
          "1",
          "0",
          "1",
          "0",
          "1",
          "0",
          "0",
          "0",
          "1",
          "0",
          "1",
          "0",
          "1",
          "0",
          "1",
          "0",
          "1",
          "0"
        ],
        "output": "10",
        "label": "product modulus exact coprime indicator count"
      }
    },
    {
      "id": "S078",
      "uses": [
        "G001",
        "S077"
      ],
      "before": "product modulus:complete representative range1..22;all selected coprime representatives=[1, 3, 5, 7, 9, 13, 15, 17, 19, 21]",
      "after": "phi(22)=10;representatives_1_through_n",
      "relation": "implies",
      "rule_id": "totient_complete_finite_count_not_sample_formula",
      "conditions": [],
      "explanation": {
        "zh-Hant": "全部指示量相加得到精確個數，這是指定n的完整有限計數，不從幾個樣本外推一般公式。",
        "en": "Sum every indicator for the exact count.This is complete finite counting for the supplied n,not a general formula extrapolated from samples."
      },
      "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": "S079",
      "uses": [
        "G001",
        "S078"
      ],
      "before": "totient supplied instance product of counts := mul(1, 10)",
      "after": "totient supplied instance product of counts = 10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始欄位與已核中間量精確計算，保留索引、分母、符號與運算角色。",
        "en": "Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and operation roles."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 totient supplied instance product of counts，不以小數近似替代。",
          "en": "Recompute totient supplied instance product of counts with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "10"
        ],
        "output": "10",
        "label": "totient supplied instance product of counts"
      }
    },
    {
      "id": "S080",
      "uses": [
        "G001",
        "S079"
      ],
      "before": "original coprime inputs=[2, 11];gcd=1;product modulus=22;exact totients=[1, 10, 10]",
      "after": "phi(m)*phi(n)=10;same supplied-instance equality=是;instance_only_with_coprimality",
      "relation": "implies",
      "rule_id": "totient_coprime_instance_not_unrestricted_multiplicativity_proof",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先核互質且三個模數都在來源允許域，再完整計三個φ並比較。這只核當前實例；未互質不套用本題協定，也不把數值相等當一般乘法性定理的證明。",
        "en": "First verify coprimality and all three allowed moduli,then count the three totients completely and compare.This checks the supplied instance only.Non-coprime inputs do not qualify,and numeric agreement is not a proof of the universal multiplicativity theorem."
      },
      "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": [
      2,
      11,
      1,
      10,
      10,
      10,
      true,
      "instance_only_with_coprimality"
    ]
  },
  "conclusion": {
    "canonical_value": "(2, 11, 1, 10, 10, 10, 是, instance_only_with_coprimality)",
    "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-20261101-r06-euler-totient-function-foundations",
      "batch_path": "public/data/questions/batches/batch-20261014-advance-20261101-r06-euler-totient-function-foundations/questions.jsonl",
      "batch_sha256": "be9ba3dbd5d7a05b739ac7f3c674100eeccfbe535f78bd6c97472630bc1d0d08",
      "original_line": 80,
      "original_line_sha256": "731e2b9f07b63ed78e11183f71d8b34c6dffbc6ca1f4356b128986185efc1d93",
      "question_compact_sha256": "f0912a061666c95346617ae069f296c6b057c2a4a9dda97fd0740a94d168024b",
      "english_row_compact_sha256": "b266946ef922cc253e4698578a2b928f89ac7fad24413c36166901fe4c68ce04",
      "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261101-r06-euler-totient-function-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261101-r06-euler-totient-function-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-euler-totient-function": {
        "sha256": "2f96ece219d918a70b350ee38d51917541627e34c40240cf2b2cfa6d4ab48c31",
        "english_sha256": "55a6b7b7404726164058c3d62e026bd28463e20a31eabb18f1a001b38bc4907d",
        "original_path": "public/data/mko/euler-totient-function.json",
        "english_path": "public/data/i18n/editions/mko-euler-totient-function.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "fe4c48735753bdd58c10db9b2ca3a2a3baeb31d0424c3fc09dea62317185ddae",
    "original_candidate_sha256": "84516a4b77b05c5f8649f4a54546ba689f0be603c3397326e0c91ce1f1eed464",
    "original_candidate_line_sha256": "84516a4b77b05c5f8649f4a54546ba689f0be603c3397326e0c91ce1f1eed464"
  },
  "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-20261026-007-p33",
    "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": "d64b4e08bda1c25dd393b7c7e4140a8734610a9b84ae47e6f0adcff7f30a0b8e",
      "question_first_witness_sha256": "790fd6623c04ba52a6ffb0a95cea52b532a02b23bb9453926d2d9f7a825fdec8"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 80,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "c4e555b0eead0337f0b731d1d7cf61b3f2306ed7d36697793cfb4cc22e798e2e",
      "arithmetic_steps": 5
    },
    "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."
  }
}