{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0095",
  "pilot_number": "R0043-22145",
  "number": 22145,
  "family_id": "qf-mod41-mobinv-incomplete-divisor-boundary",
  "title": {
    "zh-Hant": "第 22145 題：有限因數格上的Möbius反演",
    "en": "Question 22145: An Omitted Divisor Must Be a Discriminative Missing-Term Witness"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0095",
    "batch_id": "batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-mobius-inversion-finite-divisors"
    ],
    "title_zh": "有限因數格上的Möbius反演",
    "stem_zh": "positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-incomplete-divisor-boundary；資料={\"n\":462,\"a\":20,\"b\":1,\"omitted_divisor\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(462, [1, 2, 3, 6, 7, 11, 14, 21, 22, 33, 42, 66, 77, 154, 231, 462], 1, [2, 3, 6, 7, 11, 14, 21, 22, 33, 42, 66, 77, 154, 231, 462], 9241, -13815, 9241, 否, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)",
      "explanation_zh": "Wave6第四層核對=(462, [1, 2, 3, 6, 7, 11, 14, 21, 22, 33, 42, 66, 77, 154, 231, 462], 1, [2, 3, 6, 7, 11, 14, 21, 22, 33, 42, 66, 77, 154, 231, 462], 9241, -13815, 9241, 否, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)；依完整明示資料獨立重算；使用F(n)=sum_{d|n}f(d)與f(n)=sum_{d|n}mu(d)F(n/d)的一致indexing；不可只取prime/proper divisors或d<=n；n=1與mu(1)=1必須保留。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 3,
      "skills": [
        "有限因數格上的Möbius反演"
      ]
    },
    "variant": {
      "family_id": "qf-mod41-mobinv-incomplete-divisor-boundary",
      "seed": 202611106095,
      "parameters": {
        "n": 462,
        "a": 20,
        "b": 1,
        "omitted_divisor": 1,
        "domain": "Exact_finite_divisor_Mobius_inversion",
        "operation": "mobinv-incomplete-divisor-boundary",
        "semantic_template": "qf-mod41-mobinv-incomplete-divisor-boundary/{\"a\":20,\"b\":1,\"n\":462,\"omitted_divisor\":1}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_finite_or_symbolic_model",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-14",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0095",
    "batch_id": "batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "dddf2a7931dc250d7b88cb300d71e45113ffd09679958f4339c1fcfa7a826586",
      "family_id": "qf-mod41-mobinv-incomplete-divisor-boundary",
      "seed": 202611106095,
      "semantic_template": "qf-mod41-mobinv-incomplete-divisor-boundary/{\"a\":20,\"b\":1,\"n\":462,\"omitted_divisor\":1}"
    },
    "fields": {
      "title": "An Omitted Divisor Must Be a Discriminative Missing-Term Witness",
      "stem": "Use positive integer n=462, exact integer arithmetic function f(k)=a*k+b with a=20, b=1. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Omit divisor=1. Give (n, all_divisors, omitted_divisor, partial_divisors, complete_value, partial_value, f_n, incomplete_certificate, boundary).",
      "answer": {
        "kind": "expression",
        "value": "(462, [1, 2, 3, 6, 7, 11, 14, 21, 22, 33, 42, 66, 77, 154, 231, 462], 1, [2, 3, 6, 7, 11, 14, 21, 22, 33, 42, 66, 77, 154, 231, 462], 9241, -13815, 9241, No, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)",
        "explanation": "Require the omitted value actually to divide n and the incomplete inverse to differ from f(n). Retain full/partial divisor lists and both sums. The fixed No refuses an incomplete certificate, not the correct complete inverse. The exact result, retaining the requested order, is (462, [1, 2, 3, 6, 7, 11, 14, 21, 22, 33, 42, 66, 77, 154, 231, 462], 1, [2, 3, 6, 7, 11, 14, 21, 22, 33, 42, 66, 77, 154, 231, 462], 9241, -13815, 9241, No, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)."
      },
      "skills": [
        "An Omitted Divisor Must Be a Discriminative Missing-Term Witness"
      ],
      "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": "positive integer domain、完整positive divisor set與完整UPF-derived Mobius values。操作=mobinv-incomplete-divisor-boundary；資料={\"n\":462,\"a\":20,\"b\":1,\"omitted_divisor\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
        "en": "Use positive integer n=462, exact integer arithmetic function f(k)=a*k+b with a=20, b=1. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Omit divisor=1. Give (n, all_divisors, omitted_divisor, partial_divisors, complete_value, partial_value, f_n, incomplete_certificate, boundary)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source operation=mobinv-incomplete-divisor-boundary;original snapshot and mathematics retained",
      "after": "explicit ordered source output fields=[n, all_divisors, omitted_divisor, partial_divisors, complete_value, partial_value, f_n, incomplete_certificate, boundary]",
      "relation": "implies",
      "rule_id": "source_ordered_interface_explicit_without_javascript",
      "conditions": [],
      "explanation": {
        "zh-Hant": "原題歷史介面標記保留，直接列來源有序欄位，不需JavaScript；MVT由原mode選欄序，不補空欄或看答案形狀猜。tag與固定No保原聲稱範圍。",
        "en": "Retain the historical interface marker and list ordered source fields without JavaScript.MVT fields come from the original mode,without padding or answer-shape inference.Tags and fixed No markers retain their claim scope."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "Mobius inversion ORIGINAL inverse:actual positive input=462;check complete integer range1..462",
      "after": "ALL positive divisors INCREASING=[1, 2, 3, 6, 7, 11, 14, 21, 22, 33, 42, 66, 77, 154, 231, 462];not only primes/proper divisors or all integers<=n",
      "relation": "implies",
      "rule_id": "mobinv_ALL_positive_divisors_complete_indexing",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整範圍查整除，保1與n；d|n不是d<=n，也不只取prime或proper divisor。",
        "en": "Check divisibility over the complete range,retaining1 and n.The relation d|n is neither d<=n nor a prime/proper-divisor restriction."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor1:actual positive input for mu=1",
      "after": "complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search",
      "relation": "implies",
      "rule_id": "mobius_complete_UPF_input_role",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先明示正在算原n還是其正因數的mu；每次需要完整prime重數，不能用找到的幾個因數猜squarefree。",
        "en": "Identify whether the input is the original n or one of its positive divisors.Every mu evaluation requires complete prime multiplicities,not a partial factor search."
      },
      "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": "Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor1:input=1;COMPLETE prime multiset=[];distinct primes=[]",
      "after": "mu=1;empty product case=是;repeated prime=否",
      "relation": "implies",
      "rule_id": "mobius_value_one_repeated_prime_or_squarefree_parity",
      "conditions": [],
      "explanation": {
        "zh-Hant": "1的空prime多重集給mu1=1；任何prime重複表示square divisor而給0；否則依不同prime個數奇偶給正負。排序不丟重數。",
        "en": "The empty prime multiset for1 gives mu1=1.A repeated prime gives a square divisor and value0;otherwise distinct-prime parity determines the sign.Sorting must not discard multiplicities."
      },
      "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": "Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor1 := div(462, 1)",
      "after": "Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor1 = 462",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor1，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "462",
          "1"
        ],
        "output": "462",
        "label": "Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor1"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/1):actual positive input=462;check complete integer range1..462",
      "after": "ALL positive divisors INCREASING=[1, 2, 3, 6, 7, 11, 14, 21, 22, 33, 42, 66, 77, 154, 231, 462];not only primes/proper divisors or all integers<=n",
      "relation": "implies",
      "rule_id": "mobinv_ALL_positive_divisors_complete_indexing",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整範圍查整除，保1與n；d|n不是d<=n，也不只取prime或proper divisor。",
        "en": "Check divisibility over the complete range,retaining1 and n.The relation d|n is neither d<=n nor a prime/proper-divisor restriction."
      },
      "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": "Mobius inversion ORIGINAL inverse F(n/1) f(1) ORIGINAL a times divisor := mul(20, 1)",
      "after": "Mobius inversion ORIGINAL inverse F(n/1) f(1) ORIGINAL a times divisor = 20",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(1) ORIGINAL a times divisor，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/1) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "20",
          "1"
        ],
        "output": "20",
        "label": "Mobius inversion ORIGINAL inverse F(n/1) f(1) ORIGINAL a times divisor"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/1) f(1) plus ORIGINAL b := sum(20, 1)",
      "after": "Mobius inversion ORIGINAL inverse F(n/1) f(1) plus ORIGINAL b = 21",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(1) plus ORIGINAL b，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/1) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "20",
          "1"
        ],
        "output": "21",
        "label": "Mobius inversion ORIGINAL inverse F(n/1) f(1) plus ORIGINAL b"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/1) f(2) ORIGINAL a times divisor := mul(20, 2)",
      "after": "Mobius inversion ORIGINAL inverse F(n/1) f(2) ORIGINAL a times divisor = 40",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(2) ORIGINAL a times divisor，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/1) f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "20",
          "2"
        ],
        "output": "40",
        "label": "Mobius inversion ORIGINAL inverse F(n/1) f(2) ORIGINAL a times divisor"
      }
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/1) f(2) plus ORIGINAL b := sum(40, 1)",
      "after": "Mobius inversion ORIGINAL inverse F(n/1) f(2) plus ORIGINAL b = 41",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(2) plus ORIGINAL b，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/1) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "40",
          "1"
        ],
        "output": "41",
        "label": "Mobius inversion ORIGINAL inverse F(n/1) f(2) plus ORIGINAL b"
      }
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/1) f(3) ORIGINAL a times divisor := mul(20, 3)",
      "after": "Mobius inversion ORIGINAL inverse F(n/1) f(3) ORIGINAL a times divisor = 60",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(3) ORIGINAL a times divisor，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/1) f(3) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "20",
          "3"
        ],
        "output": "60",
        "label": "Mobius inversion ORIGINAL inverse F(n/1) f(3) ORIGINAL a times divisor"
      }
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/1) f(3) plus ORIGINAL b := sum(60, 1)",
      "after": "Mobius inversion ORIGINAL inverse F(n/1) f(3) plus ORIGINAL b = 61",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(3) plus ORIGINAL b，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/1) f(3) plus ORIGINAL b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "60",
          "1"
        ],
        "output": "61",
        "label": "Mobius inversion ORIGINAL inverse F(n/1) f(3) plus ORIGINAL b"
      }
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/1) f(6) ORIGINAL a times divisor := mul(20, 6)",
      "after": "Mobius inversion ORIGINAL inverse F(n/1) f(6) ORIGINAL a times divisor = 120",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(6) ORIGINAL a times divisor，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/1) f(6) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "20",
          "6"
        ],
        "output": "120",
        "label": "Mobius inversion ORIGINAL inverse F(n/1) f(6) ORIGINAL a times divisor"
      }
    },
    {
      "id": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/1) f(6) plus ORIGINAL b := sum(120, 1)",
      "after": "Mobius inversion ORIGINAL inverse F(n/1) f(6) plus ORIGINAL b = 121",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(6) plus ORIGINAL b，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/1) f(6) plus ORIGINAL b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "120",
          "1"
        ],
        "output": "121",
        "label": "Mobius inversion ORIGINAL inverse F(n/1) f(6) plus ORIGINAL b"
      }
    },
    {
      "id": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/1) f(7) ORIGINAL a times divisor := mul(20, 7)",
      "after": "Mobius inversion ORIGINAL inverse F(n/1) f(7) ORIGINAL a times divisor = 140",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(7) ORIGINAL a times divisor，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/1) f(7) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "20",
          "7"
        ],
        "output": "140",
        "label": "Mobius inversion ORIGINAL inverse F(n/1) f(7) ORIGINAL a times divisor"
      }
    },
    {
      "id": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/1) f(7) plus ORIGINAL b := sum(140, 1)",
      "after": "Mobius inversion ORIGINAL inverse F(n/1) f(7) plus ORIGINAL b = 141",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(7) plus ORIGINAL b，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/1) f(7) plus ORIGINAL b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "140",
          "1"
        ],
        "output": "141",
        "label": "Mobius inversion ORIGINAL inverse F(n/1) f(7) plus ORIGINAL b"
      }
    },
    {
      "id": "S017",
      "uses": [
        "G001",
        "S016"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/1) f(11) ORIGINAL a times divisor := mul(20, 11)",
      "after": "Mobius inversion ORIGINAL inverse F(n/1) f(11) ORIGINAL a times divisor = 220",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(11) ORIGINAL a times divisor，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/1) f(11) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "20",
          "11"
        ],
        "output": "220",
        "label": "Mobius inversion ORIGINAL inverse F(n/1) f(11) ORIGINAL a times divisor"
      }
    },
    {
      "id": "S018",
      "uses": [
        "G001",
        "S017"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/1) f(11) plus ORIGINAL b := sum(220, 1)",
      "after": "Mobius inversion ORIGINAL inverse F(n/1) f(11) plus ORIGINAL b = 221",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(11) plus ORIGINAL b，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/1) f(11) plus ORIGINAL b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "220",
          "1"
        ],
        "output": "221",
        "label": "Mobius inversion ORIGINAL inverse F(n/1) f(11) plus ORIGINAL b"
      }
    },
    {
      "id": "S019",
      "uses": [
        "G001",
        "S018"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/1) f(14) ORIGINAL a times divisor := mul(20, 14)",
      "after": "Mobius inversion ORIGINAL inverse F(n/1) f(14) ORIGINAL a times divisor = 280",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(14) ORIGINAL a times divisor，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/1) f(14) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "20",
          "14"
        ],
        "output": "280",
        "label": "Mobius inversion ORIGINAL inverse F(n/1) f(14) ORIGINAL a times divisor"
      }
    },
    {
      "id": "S020",
      "uses": [
        "G001",
        "S019"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/1) f(14) plus ORIGINAL b := sum(280, 1)",
      "after": "Mobius inversion ORIGINAL inverse F(n/1) f(14) plus ORIGINAL b = 281",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(14) plus ORIGINAL b，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/1) f(14) plus ORIGINAL b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "280",
          "1"
        ],
        "output": "281",
        "label": "Mobius inversion ORIGINAL inverse F(n/1) f(14) plus ORIGINAL b"
      }
    },
    {
      "id": "S021",
      "uses": [
        "G001",
        "S020"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/1) f(21) ORIGINAL a times divisor := mul(20, 21)",
      "after": "Mobius inversion ORIGINAL inverse F(n/1) f(21) ORIGINAL a times divisor = 420",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(21) ORIGINAL a times divisor，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/1) f(21) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "20",
          "21"
        ],
        "output": "420",
        "label": "Mobius inversion ORIGINAL inverse F(n/1) f(21) ORIGINAL a times divisor"
      }
    },
    {
      "id": "S022",
      "uses": [
        "G001",
        "S021"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/1) f(21) plus ORIGINAL b := sum(420, 1)",
      "after": "Mobius inversion ORIGINAL inverse F(n/1) f(21) plus ORIGINAL b = 421",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(21) plus ORIGINAL b，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/1) f(21) plus ORIGINAL b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "420",
          "1"
        ],
        "output": "421",
        "label": "Mobius inversion ORIGINAL inverse F(n/1) f(21) plus ORIGINAL b"
      }
    },
    {
      "id": "S023",
      "uses": [
        "G001",
        "S022"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/1) f(22) ORIGINAL a times divisor := mul(20, 22)",
      "after": "Mobius inversion ORIGINAL inverse F(n/1) f(22) ORIGINAL a times divisor = 440",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(22) ORIGINAL a times divisor，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/1) f(22) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "20",
          "22"
        ],
        "output": "440",
        "label": "Mobius inversion ORIGINAL inverse F(n/1) f(22) ORIGINAL a times divisor"
      }
    },
    {
      "id": "S024",
      "uses": [
        "G001",
        "S023"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/1) f(22) plus ORIGINAL b := sum(440, 1)",
      "after": "Mobius inversion ORIGINAL inverse F(n/1) f(22) plus ORIGINAL b = 441",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(22) plus ORIGINAL b，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/1) f(22) plus ORIGINAL b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "440",
          "1"
        ],
        "output": "441",
        "label": "Mobius inversion ORIGINAL inverse F(n/1) f(22) plus ORIGINAL b"
      }
    },
    {
      "id": "S025",
      "uses": [
        "G001",
        "S024"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/1) f(33) ORIGINAL a times divisor := mul(20, 33)",
      "after": "Mobius inversion ORIGINAL inverse F(n/1) f(33) ORIGINAL a times divisor = 660",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(33) ORIGINAL a times divisor，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/1) f(33) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "20",
          "33"
        ],
        "output": "660",
        "label": "Mobius inversion ORIGINAL inverse F(n/1) f(33) ORIGINAL a times divisor"
      }
    },
    {
      "id": "S026",
      "uses": [
        "G001",
        "S025"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/1) f(33) plus ORIGINAL b := sum(660, 1)",
      "after": "Mobius inversion ORIGINAL inverse F(n/1) f(33) plus ORIGINAL b = 661",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(33) plus ORIGINAL b，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/1) f(33) plus ORIGINAL b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "660",
          "1"
        ],
        "output": "661",
        "label": "Mobius inversion ORIGINAL inverse F(n/1) f(33) plus ORIGINAL b"
      }
    },
    {
      "id": "S027",
      "uses": [
        "G001",
        "S026"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/1) f(42) ORIGINAL a times divisor := mul(20, 42)",
      "after": "Mobius inversion ORIGINAL inverse F(n/1) f(42) ORIGINAL a times divisor = 840",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(42) ORIGINAL a times divisor，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/1) f(42) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "20",
          "42"
        ],
        "output": "840",
        "label": "Mobius inversion ORIGINAL inverse F(n/1) f(42) ORIGINAL a times divisor"
      }
    },
    {
      "id": "S028",
      "uses": [
        "G001",
        "S027"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/1) f(42) plus ORIGINAL b := sum(840, 1)",
      "after": "Mobius inversion ORIGINAL inverse F(n/1) f(42) plus ORIGINAL b = 841",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(42) plus ORIGINAL b，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/1) f(42) plus ORIGINAL b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "840",
          "1"
        ],
        "output": "841",
        "label": "Mobius inversion ORIGINAL inverse F(n/1) f(42) plus ORIGINAL b"
      }
    },
    {
      "id": "S029",
      "uses": [
        "G001",
        "S028"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/1) f(66) ORIGINAL a times divisor := mul(20, 66)",
      "after": "Mobius inversion ORIGINAL inverse F(n/1) f(66) ORIGINAL a times divisor = 1320",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(66) ORIGINAL a times divisor，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/1) f(66) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "20",
          "66"
        ],
        "output": "1320",
        "label": "Mobius inversion ORIGINAL inverse F(n/1) f(66) ORIGINAL a times divisor"
      }
    },
    {
      "id": "S030",
      "uses": [
        "G001",
        "S029"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/1) f(66) plus ORIGINAL b := sum(1320, 1)",
      "after": "Mobius inversion ORIGINAL inverse F(n/1) f(66) plus ORIGINAL b = 1321",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(66) plus ORIGINAL b，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/1) f(66) plus ORIGINAL b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1320",
          "1"
        ],
        "output": "1321",
        "label": "Mobius inversion ORIGINAL inverse F(n/1) f(66) plus ORIGINAL b"
      }
    },
    {
      "id": "S031",
      "uses": [
        "G001",
        "S030"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/1) f(77) ORIGINAL a times divisor := mul(20, 77)",
      "after": "Mobius inversion ORIGINAL inverse F(n/1) f(77) ORIGINAL a times divisor = 1540",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(77) ORIGINAL a times divisor，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/1) f(77) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "20",
          "77"
        ],
        "output": "1540",
        "label": "Mobius inversion ORIGINAL inverse F(n/1) f(77) ORIGINAL a times divisor"
      }
    },
    {
      "id": "S032",
      "uses": [
        "G001",
        "S031"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/1) f(77) plus ORIGINAL b := sum(1540, 1)",
      "after": "Mobius inversion ORIGINAL inverse F(n/1) f(77) plus ORIGINAL b = 1541",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(77) plus ORIGINAL b，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/1) f(77) plus ORIGINAL b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1540",
          "1"
        ],
        "output": "1541",
        "label": "Mobius inversion ORIGINAL inverse F(n/1) f(77) plus ORIGINAL b"
      }
    },
    {
      "id": "S033",
      "uses": [
        "G001",
        "S032"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/1) f(154) ORIGINAL a times divisor := mul(20, 154)",
      "after": "Mobius inversion ORIGINAL inverse F(n/1) f(154) ORIGINAL a times divisor = 3080",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(154) ORIGINAL a times divisor，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/1) f(154) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "20",
          "154"
        ],
        "output": "3080",
        "label": "Mobius inversion ORIGINAL inverse F(n/1) f(154) ORIGINAL a times divisor"
      }
    },
    {
      "id": "S034",
      "uses": [
        "G001",
        "S033"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/1) f(154) plus ORIGINAL b := sum(3080, 1)",
      "after": "Mobius inversion ORIGINAL inverse F(n/1) f(154) plus ORIGINAL b = 3081",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(154) plus ORIGINAL b，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/1) f(154) plus ORIGINAL b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "3080",
          "1"
        ],
        "output": "3081",
        "label": "Mobius inversion ORIGINAL inverse F(n/1) f(154) plus ORIGINAL b"
      }
    },
    {
      "id": "S035",
      "uses": [
        "G001",
        "S034"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/1) f(231) ORIGINAL a times divisor := mul(20, 231)",
      "after": "Mobius inversion ORIGINAL inverse F(n/1) f(231) ORIGINAL a times divisor = 4620",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(231) ORIGINAL a times divisor，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/1) f(231) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "20",
          "231"
        ],
        "output": "4620",
        "label": "Mobius inversion ORIGINAL inverse F(n/1) f(231) ORIGINAL a times divisor"
      }
    },
    {
      "id": "S036",
      "uses": [
        "G001",
        "S035"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/1) f(231) plus ORIGINAL b := sum(4620, 1)",
      "after": "Mobius inversion ORIGINAL inverse F(n/1) f(231) plus ORIGINAL b = 4621",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(231) plus ORIGINAL b，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/1) f(231) plus ORIGINAL b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "4620",
          "1"
        ],
        "output": "4621",
        "label": "Mobius inversion ORIGINAL inverse F(n/1) f(231) plus ORIGINAL b"
      }
    },
    {
      "id": "S037",
      "uses": [
        "G001",
        "S036"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/1) f(462) ORIGINAL a times divisor := mul(20, 462)",
      "after": "Mobius inversion ORIGINAL inverse F(n/1) f(462) ORIGINAL a times divisor = 9240",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(462) ORIGINAL a times divisor，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/1) f(462) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "20",
          "462"
        ],
        "output": "9240",
        "label": "Mobius inversion ORIGINAL inverse F(n/1) f(462) ORIGINAL a times divisor"
      }
    },
    {
      "id": "S038",
      "uses": [
        "G001",
        "S037"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/1) f(462) plus ORIGINAL b := sum(9240, 1)",
      "after": "Mobius inversion ORIGINAL inverse F(n/1) f(462) plus ORIGINAL b = 9241",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) f(462) plus ORIGINAL b，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/1) f(462) plus ORIGINAL b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "9240",
          "1"
        ],
        "output": "9241",
        "label": "Mobius inversion ORIGINAL inverse F(n/1) f(462) plus ORIGINAL b"
      }
    },
    {
      "id": "S039",
      "uses": [
        "G001",
        "S038"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/1) full forward divisor sum := sum(21, 41, 61, 121, 141, 221, 281, 421, 441, 661, 841, 1321, 1541, 3081, 4621, 9241)",
      "after": "Mobius inversion ORIGINAL inverse F(n/1) full forward divisor sum = 23056",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/1) full forward divisor sum，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/1) full forward divisor sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "21",
          "41",
          "61",
          "121",
          "141",
          "221",
          "281",
          "421",
          "441",
          "661",
          "841",
          "1321",
          "1541",
          "3081",
          "4621",
          "9241"
        ],
        "output": "23056",
        "label": "Mobius inversion ORIGINAL inverse F(n/1) full forward divisor sum"
      }
    },
    {
      "id": "S040",
      "uses": [
        "G001",
        "S039"
      ],
      "before": "Mobius inversion ORIGINAL inverse mu(1) times SAME F(n/1) := mul(1, 23056)",
      "after": "Mobius inversion ORIGINAL inverse mu(1) times SAME F(n/1) = 23056",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse mu(1) times SAME F(n/1)，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse mu(1) times SAME F(n/1) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "23056"
        ],
        "output": "23056",
        "label": "Mobius inversion ORIGINAL inverse mu(1) times SAME F(n/1)"
      }
    },
    {
      "id": "S041",
      "uses": [
        "G001",
        "S040"
      ],
      "before": "Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor2:actual positive input for mu=2",
      "after": "complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search",
      "relation": "implies",
      "rule_id": "mobius_complete_UPF_input_role",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先明示正在算原n還是其正因數的mu；每次需要完整prime重數，不能用找到的幾個因數猜squarefree。",
        "en": "Identify whether the input is the original n or one of its positive divisors.Every mu evaluation requires complete prime multiplicities,not a partial factor search."
      },
      "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": "ORIGINAL positive integer absolute magnitude=2>1",
      "after": "initialize remaining=n,prime multiset empty;invariant product(extracted)*remaining=n;every smaller trial divisor has been exhausted",
      "relation": "implies",
      "rule_id": "upf_factorization_remaining_product_invariant",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每次只除真正整除的d，並以已窮盡更小因數維持d為prime的理由，不先假定原n為prime。",
        "en": "Divide only by an actual divisor and exhaust smaller candidates,which justifies primality of each extracted divisor without assuming n is prime."
      },
      "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": "upf canonical complete product factor0 := mul(1, 2)",
      "after": "upf canonical complete product factor0 = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。",
        "en": "Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 upf canonical complete product factor0，不以小數近似替代。",
          "en": "Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "2"
        ],
        "output": "2",
        "label": "upf canonical complete product factor0"
      }
    },
    {
      "id": "S044",
      "uses": [
        "G001",
        "S043"
      ],
      "before": "exhausted trial divisors below2;terminal remaining=2;d*d>remaining;ORIGINAL n=2",
      "after": "complete increasing prime multiset=[2];full product=2;a composite terminal remainder would have an already tested smaller divisor",
      "relation": "implies",
      "rule_id": "upf_final_prime_remainder_complete_existence_instance",
      "conditions": [],
      "explanation": {
        "zh-Hant": "若remaining>1仍合數，其最小prime因數會在已驗範圍，與除盡不變式矛盾，所以補入的末項為prime。這證本例完整分解；一般存在也可對n作強歸納，合數拆成較小正因數後遞歸。",
        "en": "If a terminal remainder greater than1 were composite,its least prime factor would already have been exhausted,a contradiction.The final entry is therefore prime.This certifies this instance;general existence follows by strong induction,recursively splitting a composite into smaller positive factors."
      },
      "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": "Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor2:input=2;COMPLETE prime multiset=[2];distinct primes=[2]",
      "after": "mu=-1;empty product case=否;repeated prime=否",
      "relation": "implies",
      "rule_id": "mobius_value_one_repeated_prime_or_squarefree_parity",
      "conditions": [],
      "explanation": {
        "zh-Hant": "1的空prime多重集給mu1=1；任何prime重複表示square divisor而給0；否則依不同prime個數奇偶給正負。排序不丟重數。",
        "en": "The empty prime multiset for1 gives mu1=1.A repeated prime gives a square divisor and value0;otherwise distinct-prime parity determines the sign.Sorting must not discard multiplicities."
      },
      "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": "Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor2 := div(462, 2)",
      "after": "Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor2 = 231",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor2，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "462",
          "2"
        ],
        "output": "231",
        "label": "Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor2"
      }
    },
    {
      "id": "S047",
      "uses": [
        "G001",
        "S046"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/2):actual positive input=231;check complete integer range1..231",
      "after": "ALL positive divisors INCREASING=[1, 3, 7, 11, 21, 33, 77, 231];not only primes/proper divisors or all integers<=n",
      "relation": "implies",
      "rule_id": "mobinv_ALL_positive_divisors_complete_indexing",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整範圍查整除，保1與n；d|n不是d<=n，也不只取prime或proper divisor。",
        "en": "Check divisibility over the complete range,retaining1 and n.The relation d|n is neither d<=n nor a prime/proper-divisor restriction."
      },
      "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": "Mobius inversion ORIGINAL inverse F(n/2) f(1) ORIGINAL a times divisor := mul(20, 1)",
      "after": "Mobius inversion ORIGINAL inverse F(n/2) f(1) ORIGINAL a times divisor = 20",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/2) f(1) ORIGINAL a times divisor，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/2) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "20",
          "1"
        ],
        "output": "20",
        "label": "Mobius inversion ORIGINAL inverse F(n/2) f(1) ORIGINAL a times divisor"
      }
    },
    {
      "id": "S049",
      "uses": [
        "G001",
        "S048"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/2) f(1) plus ORIGINAL b := sum(20, 1)",
      "after": "Mobius inversion ORIGINAL inverse F(n/2) f(1) plus ORIGINAL b = 21",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/2) f(1) plus ORIGINAL b，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/2) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "20",
          "1"
        ],
        "output": "21",
        "label": "Mobius inversion ORIGINAL inverse F(n/2) f(1) plus ORIGINAL b"
      }
    },
    {
      "id": "S050",
      "uses": [
        "G001",
        "S049"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/2) f(3) ORIGINAL a times divisor := mul(20, 3)",
      "after": "Mobius inversion ORIGINAL inverse F(n/2) f(3) ORIGINAL a times divisor = 60",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/2) f(3) ORIGINAL a times divisor，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/2) f(3) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "20",
          "3"
        ],
        "output": "60",
        "label": "Mobius inversion ORIGINAL inverse F(n/2) f(3) ORIGINAL a times divisor"
      }
    },
    {
      "id": "S051",
      "uses": [
        "G001",
        "S050"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/2) f(3) plus ORIGINAL b := sum(60, 1)",
      "after": "Mobius inversion ORIGINAL inverse F(n/2) f(3) plus ORIGINAL b = 61",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/2) f(3) plus ORIGINAL b，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/2) f(3) plus ORIGINAL b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "60",
          "1"
        ],
        "output": "61",
        "label": "Mobius inversion ORIGINAL inverse F(n/2) f(3) plus ORIGINAL b"
      }
    },
    {
      "id": "S052",
      "uses": [
        "G001",
        "S051"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/2) f(7) ORIGINAL a times divisor := mul(20, 7)",
      "after": "Mobius inversion ORIGINAL inverse F(n/2) f(7) ORIGINAL a times divisor = 140",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/2) f(7) ORIGINAL a times divisor，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/2) f(7) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "20",
          "7"
        ],
        "output": "140",
        "label": "Mobius inversion ORIGINAL inverse F(n/2) f(7) ORIGINAL a times divisor"
      }
    },
    {
      "id": "S053",
      "uses": [
        "G001",
        "S052"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/2) f(7) plus ORIGINAL b := sum(140, 1)",
      "after": "Mobius inversion ORIGINAL inverse F(n/2) f(7) plus ORIGINAL b = 141",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/2) f(7) plus ORIGINAL b，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/2) f(7) plus ORIGINAL b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "140",
          "1"
        ],
        "output": "141",
        "label": "Mobius inversion ORIGINAL inverse F(n/2) f(7) plus ORIGINAL b"
      }
    },
    {
      "id": "S054",
      "uses": [
        "G001",
        "S053"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/2) f(11) ORIGINAL a times divisor := mul(20, 11)",
      "after": "Mobius inversion ORIGINAL inverse F(n/2) f(11) ORIGINAL a times divisor = 220",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/2) f(11) ORIGINAL a times divisor，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/2) f(11) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "20",
          "11"
        ],
        "output": "220",
        "label": "Mobius inversion ORIGINAL inverse F(n/2) f(11) ORIGINAL a times divisor"
      }
    },
    {
      "id": "S055",
      "uses": [
        "G001",
        "S054"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/2) f(11) plus ORIGINAL b := sum(220, 1)",
      "after": "Mobius inversion ORIGINAL inverse F(n/2) f(11) plus ORIGINAL b = 221",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/2) f(11) plus ORIGINAL b，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/2) f(11) plus ORIGINAL b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "220",
          "1"
        ],
        "output": "221",
        "label": "Mobius inversion ORIGINAL inverse F(n/2) f(11) plus ORIGINAL b"
      }
    },
    {
      "id": "S056",
      "uses": [
        "G001",
        "S055"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/2) f(21) ORIGINAL a times divisor := mul(20, 21)",
      "after": "Mobius inversion ORIGINAL inverse F(n/2) f(21) ORIGINAL a times divisor = 420",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/2) f(21) ORIGINAL a times divisor，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/2) f(21) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "20",
          "21"
        ],
        "output": "420",
        "label": "Mobius inversion ORIGINAL inverse F(n/2) f(21) ORIGINAL a times divisor"
      }
    },
    {
      "id": "S057",
      "uses": [
        "G001",
        "S056"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/2) f(21) plus ORIGINAL b := sum(420, 1)",
      "after": "Mobius inversion ORIGINAL inverse F(n/2) f(21) plus ORIGINAL b = 421",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/2) f(21) plus ORIGINAL b，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/2) f(21) plus ORIGINAL b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "420",
          "1"
        ],
        "output": "421",
        "label": "Mobius inversion ORIGINAL inverse F(n/2) f(21) plus ORIGINAL b"
      }
    },
    {
      "id": "S058",
      "uses": [
        "G001",
        "S057"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/2) f(33) ORIGINAL a times divisor := mul(20, 33)",
      "after": "Mobius inversion ORIGINAL inverse F(n/2) f(33) ORIGINAL a times divisor = 660",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/2) f(33) ORIGINAL a times divisor，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/2) f(33) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "20",
          "33"
        ],
        "output": "660",
        "label": "Mobius inversion ORIGINAL inverse F(n/2) f(33) ORIGINAL a times divisor"
      }
    },
    {
      "id": "S059",
      "uses": [
        "G001",
        "S058"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/2) f(33) plus ORIGINAL b := sum(660, 1)",
      "after": "Mobius inversion ORIGINAL inverse F(n/2) f(33) plus ORIGINAL b = 661",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/2) f(33) plus ORIGINAL b，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/2) f(33) plus ORIGINAL b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "660",
          "1"
        ],
        "output": "661",
        "label": "Mobius inversion ORIGINAL inverse F(n/2) f(33) plus ORIGINAL b"
      }
    },
    {
      "id": "S060",
      "uses": [
        "G001",
        "S059"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/2) f(77) ORIGINAL a times divisor := mul(20, 77)",
      "after": "Mobius inversion ORIGINAL inverse F(n/2) f(77) ORIGINAL a times divisor = 1540",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/2) f(77) ORIGINAL a times divisor，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/2) f(77) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "20",
          "77"
        ],
        "output": "1540",
        "label": "Mobius inversion ORIGINAL inverse F(n/2) f(77) ORIGINAL a times divisor"
      }
    },
    {
      "id": "S061",
      "uses": [
        "G001",
        "S060"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/2) f(77) plus ORIGINAL b := sum(1540, 1)",
      "after": "Mobius inversion ORIGINAL inverse F(n/2) f(77) plus ORIGINAL b = 1541",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/2) f(77) plus ORIGINAL b，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/2) f(77) plus ORIGINAL b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1540",
          "1"
        ],
        "output": "1541",
        "label": "Mobius inversion ORIGINAL inverse F(n/2) f(77) plus ORIGINAL b"
      }
    },
    {
      "id": "S062",
      "uses": [
        "G001",
        "S061"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/2) f(231) ORIGINAL a times divisor := mul(20, 231)",
      "after": "Mobius inversion ORIGINAL inverse F(n/2) f(231) ORIGINAL a times divisor = 4620",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/2) f(231) ORIGINAL a times divisor，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/2) f(231) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "20",
          "231"
        ],
        "output": "4620",
        "label": "Mobius inversion ORIGINAL inverse F(n/2) f(231) ORIGINAL a times divisor"
      }
    },
    {
      "id": "S063",
      "uses": [
        "G001",
        "S062"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/2) f(231) plus ORIGINAL b := sum(4620, 1)",
      "after": "Mobius inversion ORIGINAL inverse F(n/2) f(231) plus ORIGINAL b = 4621",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/2) f(231) plus ORIGINAL b，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/2) f(231) plus ORIGINAL b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "4620",
          "1"
        ],
        "output": "4621",
        "label": "Mobius inversion ORIGINAL inverse F(n/2) f(231) plus ORIGINAL b"
      }
    },
    {
      "id": "S064",
      "uses": [
        "G001",
        "S063"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/2) full forward divisor sum := sum(21, 61, 141, 221, 421, 661, 1541, 4621)",
      "after": "Mobius inversion ORIGINAL inverse F(n/2) full forward divisor sum = 7688",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/2) full forward divisor sum，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/2) full forward divisor sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "21",
          "61",
          "141",
          "221",
          "421",
          "661",
          "1541",
          "4621"
        ],
        "output": "7688",
        "label": "Mobius inversion ORIGINAL inverse F(n/2) full forward divisor sum"
      }
    },
    {
      "id": "S065",
      "uses": [
        "G001",
        "S064"
      ],
      "before": "Mobius inversion ORIGINAL inverse mu(2) times SAME F(n/2) := mul(-1, 7688)",
      "after": "Mobius inversion ORIGINAL inverse mu(2) times SAME F(n/2) = -7688",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse mu(2) times SAME F(n/2)，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse mu(2) times SAME F(n/2) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-1",
          "7688"
        ],
        "output": "-7688",
        "label": "Mobius inversion ORIGINAL inverse mu(2) times SAME F(n/2)"
      }
    },
    {
      "id": "S066",
      "uses": [
        "G001",
        "S065"
      ],
      "before": "Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor3:actual positive input for mu=3",
      "after": "complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search",
      "relation": "implies",
      "rule_id": "mobius_complete_UPF_input_role",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先明示正在算原n還是其正因數的mu；每次需要完整prime重數，不能用找到的幾個因數猜squarefree。",
        "en": "Identify whether the input is the original n or one of its positive divisors.Every mu evaluation requires complete prime multiplicities,not a partial factor search."
      },
      "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": "ORIGINAL positive integer absolute magnitude=3>1",
      "after": "initialize remaining=n,prime multiset empty;invariant product(extracted)*remaining=n;every smaller trial divisor has been exhausted",
      "relation": "implies",
      "rule_id": "upf_factorization_remaining_product_invariant",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每次只除真正整除的d，並以已窮盡更小因數維持d為prime的理由，不先假定原n為prime。",
        "en": "Divide only by an actual divisor and exhaust smaller candidates,which justifies primality of each extracted divisor without assuming n is prime."
      },
      "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": "upf canonical complete product factor0 := mul(1, 3)",
      "after": "upf canonical complete product factor0 = 3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。",
        "en": "Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 upf canonical complete product factor0，不以小數近似替代。",
          "en": "Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "3"
        ],
        "output": "3",
        "label": "upf canonical complete product factor0"
      }
    },
    {
      "id": "S069",
      "uses": [
        "G001",
        "S068"
      ],
      "before": "exhausted trial divisors below2;terminal remaining=3;d*d>remaining;ORIGINAL n=3",
      "after": "complete increasing prime multiset=[3];full product=3;a composite terminal remainder would have an already tested smaller divisor",
      "relation": "implies",
      "rule_id": "upf_final_prime_remainder_complete_existence_instance",
      "conditions": [],
      "explanation": {
        "zh-Hant": "若remaining>1仍合數，其最小prime因數會在已驗範圍，與除盡不變式矛盾，所以補入的末項為prime。這證本例完整分解；一般存在也可對n作強歸納，合數拆成較小正因數後遞歸。",
        "en": "If a terminal remainder greater than1 were composite,its least prime factor would already have been exhausted,a contradiction.The final entry is therefore prime.This certifies this instance;general existence follows by strong induction,recursively splitting a composite into smaller positive factors."
      },
      "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": "Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor3:input=3;COMPLETE prime multiset=[3];distinct primes=[3]",
      "after": "mu=-1;empty product case=否;repeated prime=否",
      "relation": "implies",
      "rule_id": "mobius_value_one_repeated_prime_or_squarefree_parity",
      "conditions": [],
      "explanation": {
        "zh-Hant": "1的空prime多重集給mu1=1；任何prime重複表示square divisor而給0；否則依不同prime個數奇偶給正負。排序不丟重數。",
        "en": "The empty prime multiset for1 gives mu1=1.A repeated prime gives a square divisor and value0;otherwise distinct-prime parity determines the sign.Sorting must not discard multiplicities."
      },
      "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": "Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor3 := div(462, 3)",
      "after": "Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor3 = 154",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor3，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor3 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "462",
          "3"
        ],
        "output": "154",
        "label": "Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor3"
      }
    },
    {
      "id": "S072",
      "uses": [
        "G001",
        "S071"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/3):actual positive input=154;check complete integer range1..154",
      "after": "ALL positive divisors INCREASING=[1, 2, 7, 11, 14, 22, 77, 154];not only primes/proper divisors or all integers<=n",
      "relation": "implies",
      "rule_id": "mobinv_ALL_positive_divisors_complete_indexing",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整範圍查整除，保1與n；d|n不是d<=n，也不只取prime或proper divisor。",
        "en": "Check divisibility over the complete range,retaining1 and n.The relation d|n is neither d<=n nor a prime/proper-divisor restriction."
      },
      "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": "Mobius inversion ORIGINAL inverse F(n/3) f(1) ORIGINAL a times divisor := mul(20, 1)",
      "after": "Mobius inversion ORIGINAL inverse F(n/3) f(1) ORIGINAL a times divisor = 20",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/3) f(1) ORIGINAL a times divisor，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/3) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "20",
          "1"
        ],
        "output": "20",
        "label": "Mobius inversion ORIGINAL inverse F(n/3) f(1) ORIGINAL a times divisor"
      }
    },
    {
      "id": "S074",
      "uses": [
        "G001",
        "S073"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/3) f(1) plus ORIGINAL b := sum(20, 1)",
      "after": "Mobius inversion ORIGINAL inverse F(n/3) f(1) plus ORIGINAL b = 21",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/3) f(1) plus ORIGINAL b，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/3) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "20",
          "1"
        ],
        "output": "21",
        "label": "Mobius inversion ORIGINAL inverse F(n/3) f(1) plus ORIGINAL b"
      }
    },
    {
      "id": "S075",
      "uses": [
        "G001",
        "S074"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/3) f(2) ORIGINAL a times divisor := mul(20, 2)",
      "after": "Mobius inversion ORIGINAL inverse F(n/3) f(2) ORIGINAL a times divisor = 40",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/3) f(2) ORIGINAL a times divisor，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/3) f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "20",
          "2"
        ],
        "output": "40",
        "label": "Mobius inversion ORIGINAL inverse F(n/3) f(2) ORIGINAL a times divisor"
      }
    },
    {
      "id": "S076",
      "uses": [
        "G001",
        "S075"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/3) f(2) plus ORIGINAL b := sum(40, 1)",
      "after": "Mobius inversion ORIGINAL inverse F(n/3) f(2) plus ORIGINAL b = 41",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/3) f(2) plus ORIGINAL b，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/3) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "40",
          "1"
        ],
        "output": "41",
        "label": "Mobius inversion ORIGINAL inverse F(n/3) f(2) plus ORIGINAL b"
      }
    },
    {
      "id": "S077",
      "uses": [
        "G001",
        "S076"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/3) f(7) ORIGINAL a times divisor := mul(20, 7)",
      "after": "Mobius inversion ORIGINAL inverse F(n/3) f(7) ORIGINAL a times divisor = 140",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/3) f(7) ORIGINAL a times divisor，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/3) f(7) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "20",
          "7"
        ],
        "output": "140",
        "label": "Mobius inversion ORIGINAL inverse F(n/3) f(7) ORIGINAL a times divisor"
      }
    },
    {
      "id": "S078",
      "uses": [
        "G001",
        "S077"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/3) f(7) plus ORIGINAL b := sum(140, 1)",
      "after": "Mobius inversion ORIGINAL inverse F(n/3) f(7) plus ORIGINAL b = 141",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/3) f(7) plus ORIGINAL b，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/3) f(7) plus ORIGINAL b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "140",
          "1"
        ],
        "output": "141",
        "label": "Mobius inversion ORIGINAL inverse F(n/3) f(7) plus ORIGINAL b"
      }
    },
    {
      "id": "S079",
      "uses": [
        "G001",
        "S078"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/3) f(11) ORIGINAL a times divisor := mul(20, 11)",
      "after": "Mobius inversion ORIGINAL inverse F(n/3) f(11) ORIGINAL a times divisor = 220",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/3) f(11) ORIGINAL a times divisor，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/3) f(11) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "20",
          "11"
        ],
        "output": "220",
        "label": "Mobius inversion ORIGINAL inverse F(n/3) f(11) ORIGINAL a times divisor"
      }
    },
    {
      "id": "S080",
      "uses": [
        "G001",
        "S079"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/3) f(11) plus ORIGINAL b := sum(220, 1)",
      "after": "Mobius inversion ORIGINAL inverse F(n/3) f(11) plus ORIGINAL b = 221",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/3) f(11) plus ORIGINAL b，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/3) f(11) plus ORIGINAL b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "220",
          "1"
        ],
        "output": "221",
        "label": "Mobius inversion ORIGINAL inverse F(n/3) f(11) plus ORIGINAL b"
      }
    },
    {
      "id": "S081",
      "uses": [
        "G001",
        "S080"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/3) f(14) ORIGINAL a times divisor := mul(20, 14)",
      "after": "Mobius inversion ORIGINAL inverse F(n/3) f(14) ORIGINAL a times divisor = 280",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/3) f(14) ORIGINAL a times divisor，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/3) f(14) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "20",
          "14"
        ],
        "output": "280",
        "label": "Mobius inversion ORIGINAL inverse F(n/3) f(14) ORIGINAL a times divisor"
      }
    },
    {
      "id": "S082",
      "uses": [
        "G001",
        "S081"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/3) f(14) plus ORIGINAL b := sum(280, 1)",
      "after": "Mobius inversion ORIGINAL inverse F(n/3) f(14) plus ORIGINAL b = 281",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/3) f(14) plus ORIGINAL b，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/3) f(14) plus ORIGINAL b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "280",
          "1"
        ],
        "output": "281",
        "label": "Mobius inversion ORIGINAL inverse F(n/3) f(14) plus ORIGINAL b"
      }
    },
    {
      "id": "S083",
      "uses": [
        "G001",
        "S082"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/3) f(22) ORIGINAL a times divisor := mul(20, 22)",
      "after": "Mobius inversion ORIGINAL inverse F(n/3) f(22) ORIGINAL a times divisor = 440",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/3) f(22) ORIGINAL a times divisor，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/3) f(22) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "20",
          "22"
        ],
        "output": "440",
        "label": "Mobius inversion ORIGINAL inverse F(n/3) f(22) ORIGINAL a times divisor"
      }
    },
    {
      "id": "S084",
      "uses": [
        "G001",
        "S083"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/3) f(22) plus ORIGINAL b := sum(440, 1)",
      "after": "Mobius inversion ORIGINAL inverse F(n/3) f(22) plus ORIGINAL b = 441",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/3) f(22) plus ORIGINAL b，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/3) f(22) plus ORIGINAL b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "440",
          "1"
        ],
        "output": "441",
        "label": "Mobius inversion ORIGINAL inverse F(n/3) f(22) plus ORIGINAL b"
      }
    },
    {
      "id": "S085",
      "uses": [
        "G001",
        "S084"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/3) f(77) ORIGINAL a times divisor := mul(20, 77)",
      "after": "Mobius inversion ORIGINAL inverse F(n/3) f(77) ORIGINAL a times divisor = 1540",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/3) f(77) ORIGINAL a times divisor，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/3) f(77) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "20",
          "77"
        ],
        "output": "1540",
        "label": "Mobius inversion ORIGINAL inverse F(n/3) f(77) ORIGINAL a times divisor"
      }
    },
    {
      "id": "S086",
      "uses": [
        "G001",
        "S085"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/3) f(77) plus ORIGINAL b := sum(1540, 1)",
      "after": "Mobius inversion ORIGINAL inverse F(n/3) f(77) plus ORIGINAL b = 1541",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/3) f(77) plus ORIGINAL b，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/3) f(77) plus ORIGINAL b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1540",
          "1"
        ],
        "output": "1541",
        "label": "Mobius inversion ORIGINAL inverse F(n/3) f(77) plus ORIGINAL b"
      }
    },
    {
      "id": "S087",
      "uses": [
        "G001",
        "S086"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/3) f(154) ORIGINAL a times divisor := mul(20, 154)",
      "after": "Mobius inversion ORIGINAL inverse F(n/3) f(154) ORIGINAL a times divisor = 3080",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/3) f(154) ORIGINAL a times divisor，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/3) f(154) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "20",
          "154"
        ],
        "output": "3080",
        "label": "Mobius inversion ORIGINAL inverse F(n/3) f(154) ORIGINAL a times divisor"
      }
    },
    {
      "id": "S088",
      "uses": [
        "G001",
        "S087"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/3) f(154) plus ORIGINAL b := sum(3080, 1)",
      "after": "Mobius inversion ORIGINAL inverse F(n/3) f(154) plus ORIGINAL b = 3081",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/3) f(154) plus ORIGINAL b，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/3) f(154) plus ORIGINAL b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "3080",
          "1"
        ],
        "output": "3081",
        "label": "Mobius inversion ORIGINAL inverse F(n/3) f(154) plus ORIGINAL b"
      }
    },
    {
      "id": "S089",
      "uses": [
        "G001",
        "S088"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/3) full forward divisor sum := sum(21, 41, 141, 221, 281, 441, 1541, 3081)",
      "after": "Mobius inversion ORIGINAL inverse F(n/3) full forward divisor sum = 5768",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/3) full forward divisor sum，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/3) full forward divisor sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "21",
          "41",
          "141",
          "221",
          "281",
          "441",
          "1541",
          "3081"
        ],
        "output": "5768",
        "label": "Mobius inversion ORIGINAL inverse F(n/3) full forward divisor sum"
      }
    },
    {
      "id": "S090",
      "uses": [
        "G001",
        "S089"
      ],
      "before": "Mobius inversion ORIGINAL inverse mu(3) times SAME F(n/3) := mul(-1, 5768)",
      "after": "Mobius inversion ORIGINAL inverse mu(3) times SAME F(n/3) = -5768",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse mu(3) times SAME F(n/3)，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse mu(3) times SAME F(n/3) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-1",
          "5768"
        ],
        "output": "-5768",
        "label": "Mobius inversion ORIGINAL inverse mu(3) times SAME F(n/3)"
      }
    },
    {
      "id": "S091",
      "uses": [
        "G001",
        "S090"
      ],
      "before": "Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor6:actual positive input for mu=6",
      "after": "complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search",
      "relation": "implies",
      "rule_id": "mobius_complete_UPF_input_role",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先明示正在算原n還是其正因數的mu；每次需要完整prime重數，不能用找到的幾個因數猜squarefree。",
        "en": "Identify whether the input is the original n or one of its positive divisors.Every mu evaluation requires complete prime multiplicities,not a partial factor search."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S092",
      "uses": [
        "G001",
        "S091"
      ],
      "before": "ORIGINAL positive integer absolute magnitude=6>1",
      "after": "initialize remaining=n,prime multiset empty;invariant product(extracted)*remaining=n;every smaller trial divisor has been exhausted",
      "relation": "implies",
      "rule_id": "upf_factorization_remaining_product_invariant",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每次只除真正整除的d，並以已窮盡更小因數維持d為prime的理由，不先假定原n為prime。",
        "en": "Divide only by an actual divisor and exhaust smaller candidates,which justifies primality of each extracted divisor without assuming n is prime."
      },
      "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": "S093",
      "uses": [
        "G001",
        "S092"
      ],
      "before": "upf extract divisor2 occurrence0 := div(6, 2)",
      "after": "upf extract divisor2 occurrence0 = 3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。",
        "en": "Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 upf extract divisor2 occurrence0，不以小數近似替代。",
          "en": "Recompute upf extract divisor2 occurrence0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "6",
          "2"
        ],
        "output": "3",
        "label": "upf extract divisor2 occurrence0"
      }
    },
    {
      "id": "S094",
      "uses": [
        "G001",
        "S093"
      ],
      "before": "next divisor=2;previous remaining=6;all smaller divisors exhausted",
      "after": "extracted multiplicity=1;next remaining=3;remaining modulo2=1;extracted multiset=[2]",
      "relation": "implies",
      "rule_id": "upf_complete_trial_division_remaining_invariant",
      "conditions": [],
      "explanation": {
        "zh-Hant": "除盡同一prime的所有重數才前進；不把n的原平方根與remaining的停止責任混淆，也不漏最後的prime remainder。",
        "en": "Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted."
      },
      "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": "S095",
      "uses": [
        "G001",
        "S094"
      ],
      "before": "upf canonical complete product factor0 := mul(1, 2)",
      "after": "upf canonical complete product factor0 = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。",
        "en": "Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 upf canonical complete product factor0，不以小數近似替代。",
          "en": "Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "2"
        ],
        "output": "2",
        "label": "upf canonical complete product factor0"
      }
    },
    {
      "id": "S096",
      "uses": [
        "G001",
        "S095"
      ],
      "before": "upf canonical complete product factor1 := mul(2, 3)",
      "after": "upf canonical complete product factor1 = 6",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。",
        "en": "Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 upf canonical complete product factor1，不以小數近似替代。",
          "en": "Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "3"
        ],
        "output": "6",
        "label": "upf canonical complete product factor1"
      }
    },
    {
      "id": "S097",
      "uses": [
        "G001",
        "S096"
      ],
      "before": "exhausted trial divisors below3;terminal remaining=3;d*d>remaining;ORIGINAL n=6",
      "after": "complete increasing prime multiset=[2, 3];full product=6;a composite terminal remainder would have an already tested smaller divisor",
      "relation": "implies",
      "rule_id": "upf_final_prime_remainder_complete_existence_instance",
      "conditions": [],
      "explanation": {
        "zh-Hant": "若remaining>1仍合數，其最小prime因數會在已驗範圍，與除盡不變式矛盾，所以補入的末項為prime。這證本例完整分解；一般存在也可對n作強歸納，合數拆成較小正因數後遞歸。",
        "en": "If a terminal remainder greater than1 were composite,its least prime factor would already have been exhausted,a contradiction.The final entry is therefore prime.This certifies this instance;general existence follows by strong induction,recursively splitting a composite into smaller positive factors."
      },
      "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": "S098",
      "uses": [
        "G001",
        "S097"
      ],
      "before": "Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor6:input=6;COMPLETE prime multiset=[2, 3];distinct primes=[2, 3]",
      "after": "mu=1;empty product case=否;repeated prime=否",
      "relation": "implies",
      "rule_id": "mobius_value_one_repeated_prime_or_squarefree_parity",
      "conditions": [],
      "explanation": {
        "zh-Hant": "1的空prime多重集給mu1=1；任何prime重複表示square divisor而給0；否則依不同prime個數奇偶給正負。排序不丟重數。",
        "en": "The empty prime multiset for1 gives mu1=1.A repeated prime gives a square divisor and value0;otherwise distinct-prime parity determines the sign.Sorting must not discard multiplicities."
      },
      "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": "S099",
      "uses": [
        "G001",
        "S098"
      ],
      "before": "Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor6 := div(462, 6)",
      "after": "Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor6 = 77",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor6，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor6 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "462",
          "6"
        ],
        "output": "77",
        "label": "Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor6"
      }
    },
    {
      "id": "S100",
      "uses": [
        "G001",
        "S099"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/6):actual positive input=77;check complete integer range1..77",
      "after": "ALL positive divisors INCREASING=[1, 7, 11, 77];not only primes/proper divisors or all integers<=n",
      "relation": "implies",
      "rule_id": "mobinv_ALL_positive_divisors_complete_indexing",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整範圍查整除，保1與n；d|n不是d<=n，也不只取prime或proper divisor。",
        "en": "Check divisibility over the complete range,retaining1 and n.The relation d|n is neither d<=n nor a prime/proper-divisor restriction."
      },
      "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": "S101",
      "uses": [
        "G001",
        "S100"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/6) f(1) ORIGINAL a times divisor := mul(20, 1)",
      "after": "Mobius inversion ORIGINAL inverse F(n/6) f(1) ORIGINAL a times divisor = 20",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/6) f(1) ORIGINAL a times divisor，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/6) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "20",
          "1"
        ],
        "output": "20",
        "label": "Mobius inversion ORIGINAL inverse F(n/6) f(1) ORIGINAL a times divisor"
      }
    },
    {
      "id": "S102",
      "uses": [
        "G001",
        "S101"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/6) f(1) plus ORIGINAL b := sum(20, 1)",
      "after": "Mobius inversion ORIGINAL inverse F(n/6) f(1) plus ORIGINAL b = 21",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/6) f(1) plus ORIGINAL b，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/6) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "20",
          "1"
        ],
        "output": "21",
        "label": "Mobius inversion ORIGINAL inverse F(n/6) f(1) plus ORIGINAL b"
      }
    },
    {
      "id": "S103",
      "uses": [
        "G001",
        "S102"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/6) f(7) ORIGINAL a times divisor := mul(20, 7)",
      "after": "Mobius inversion ORIGINAL inverse F(n/6) f(7) ORIGINAL a times divisor = 140",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/6) f(7) ORIGINAL a times divisor，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/6) f(7) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "20",
          "7"
        ],
        "output": "140",
        "label": "Mobius inversion ORIGINAL inverse F(n/6) f(7) ORIGINAL a times divisor"
      }
    },
    {
      "id": "S104",
      "uses": [
        "G001",
        "S103"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/6) f(7) plus ORIGINAL b := sum(140, 1)",
      "after": "Mobius inversion ORIGINAL inverse F(n/6) f(7) plus ORIGINAL b = 141",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/6) f(7) plus ORIGINAL b，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/6) f(7) plus ORIGINAL b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "140",
          "1"
        ],
        "output": "141",
        "label": "Mobius inversion ORIGINAL inverse F(n/6) f(7) plus ORIGINAL b"
      }
    },
    {
      "id": "S105",
      "uses": [
        "G001",
        "S104"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/6) f(11) ORIGINAL a times divisor := mul(20, 11)",
      "after": "Mobius inversion ORIGINAL inverse F(n/6) f(11) ORIGINAL a times divisor = 220",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/6) f(11) ORIGINAL a times divisor，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/6) f(11) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "20",
          "11"
        ],
        "output": "220",
        "label": "Mobius inversion ORIGINAL inverse F(n/6) f(11) ORIGINAL a times divisor"
      }
    },
    {
      "id": "S106",
      "uses": [
        "G001",
        "S105"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/6) f(11) plus ORIGINAL b := sum(220, 1)",
      "after": "Mobius inversion ORIGINAL inverse F(n/6) f(11) plus ORIGINAL b = 221",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/6) f(11) plus ORIGINAL b，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/6) f(11) plus ORIGINAL b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "220",
          "1"
        ],
        "output": "221",
        "label": "Mobius inversion ORIGINAL inverse F(n/6) f(11) plus ORIGINAL b"
      }
    },
    {
      "id": "S107",
      "uses": [
        "G001",
        "S106"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/6) f(77) ORIGINAL a times divisor := mul(20, 77)",
      "after": "Mobius inversion ORIGINAL inverse F(n/6) f(77) ORIGINAL a times divisor = 1540",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/6) f(77) ORIGINAL a times divisor，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/6) f(77) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "20",
          "77"
        ],
        "output": "1540",
        "label": "Mobius inversion ORIGINAL inverse F(n/6) f(77) ORIGINAL a times divisor"
      }
    },
    {
      "id": "S108",
      "uses": [
        "G001",
        "S107"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/6) f(77) plus ORIGINAL b := sum(1540, 1)",
      "after": "Mobius inversion ORIGINAL inverse F(n/6) f(77) plus ORIGINAL b = 1541",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/6) f(77) plus ORIGINAL b，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/6) f(77) plus ORIGINAL b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1540",
          "1"
        ],
        "output": "1541",
        "label": "Mobius inversion ORIGINAL inverse F(n/6) f(77) plus ORIGINAL b"
      }
    },
    {
      "id": "S109",
      "uses": [
        "G001",
        "S108"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/6) full forward divisor sum := sum(21, 141, 221, 1541)",
      "after": "Mobius inversion ORIGINAL inverse F(n/6) full forward divisor sum = 1924",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/6) full forward divisor sum，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/6) full forward divisor sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "21",
          "141",
          "221",
          "1541"
        ],
        "output": "1924",
        "label": "Mobius inversion ORIGINAL inverse F(n/6) full forward divisor sum"
      }
    },
    {
      "id": "S110",
      "uses": [
        "G001",
        "S109"
      ],
      "before": "Mobius inversion ORIGINAL inverse mu(6) times SAME F(n/6) := mul(1, 1924)",
      "after": "Mobius inversion ORIGINAL inverse mu(6) times SAME F(n/6) = 1924",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse mu(6) times SAME F(n/6)，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse mu(6) times SAME F(n/6) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "1924"
        ],
        "output": "1924",
        "label": "Mobius inversion ORIGINAL inverse mu(6) times SAME F(n/6)"
      }
    },
    {
      "id": "S111",
      "uses": [
        "G001",
        "S110"
      ],
      "before": "Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor7:actual positive input for mu=7",
      "after": "complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search",
      "relation": "implies",
      "rule_id": "mobius_complete_UPF_input_role",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先明示正在算原n還是其正因數的mu；每次需要完整prime重數，不能用找到的幾個因數猜squarefree。",
        "en": "Identify whether the input is the original n or one of its positive divisors.Every mu evaluation requires complete prime multiplicities,not a partial factor search."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S112",
      "uses": [
        "G001",
        "S111"
      ],
      "before": "ORIGINAL positive integer absolute magnitude=7>1",
      "after": "initialize remaining=n,prime multiset empty;invariant product(extracted)*remaining=n;every smaller trial divisor has been exhausted",
      "relation": "implies",
      "rule_id": "upf_factorization_remaining_product_invariant",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每次只除真正整除的d，並以已窮盡更小因數維持d為prime的理由，不先假定原n為prime。",
        "en": "Divide only by an actual divisor and exhaust smaller candidates,which justifies primality of each extracted divisor without assuming n is prime."
      },
      "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": "S113",
      "uses": [
        "G001",
        "S112"
      ],
      "before": "next divisor=2;previous remaining=7;all smaller divisors exhausted",
      "after": "extracted multiplicity=0;next remaining=7;remaining modulo2=1;extracted multiset=[]",
      "relation": "implies",
      "rule_id": "upf_complete_trial_division_remaining_invariant",
      "conditions": [],
      "explanation": {
        "zh-Hant": "除盡同一prime的所有重數才前進；不把n的原平方根與remaining的停止責任混淆，也不漏最後的prime remainder。",
        "en": "Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted."
      },
      "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": "S114",
      "uses": [
        "G001",
        "S113"
      ],
      "before": "upf canonical complete product factor0 := mul(1, 7)",
      "after": "upf canonical complete product factor0 = 7",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。",
        "en": "Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 upf canonical complete product factor0，不以小數近似替代。",
          "en": "Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "7"
        ],
        "output": "7",
        "label": "upf canonical complete product factor0"
      }
    },
    {
      "id": "S115",
      "uses": [
        "G001",
        "S114"
      ],
      "before": "exhausted trial divisors below3;terminal remaining=7;d*d>remaining;ORIGINAL n=7",
      "after": "complete increasing prime multiset=[7];full product=7;a composite terminal remainder would have an already tested smaller divisor",
      "relation": "implies",
      "rule_id": "upf_final_prime_remainder_complete_existence_instance",
      "conditions": [],
      "explanation": {
        "zh-Hant": "若remaining>1仍合數，其最小prime因數會在已驗範圍，與除盡不變式矛盾，所以補入的末項為prime。這證本例完整分解；一般存在也可對n作強歸納，合數拆成較小正因數後遞歸。",
        "en": "If a terminal remainder greater than1 were composite,its least prime factor would already have been exhausted,a contradiction.The final entry is therefore prime.This certifies this instance;general existence follows by strong induction,recursively splitting a composite into smaller positive factors."
      },
      "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": "S116",
      "uses": [
        "G001",
        "S115"
      ],
      "before": "Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor7:input=7;COMPLETE prime multiset=[7];distinct primes=[7]",
      "after": "mu=-1;empty product case=否;repeated prime=否",
      "relation": "implies",
      "rule_id": "mobius_value_one_repeated_prime_or_squarefree_parity",
      "conditions": [],
      "explanation": {
        "zh-Hant": "1的空prime多重集給mu1=1；任何prime重複表示square divisor而給0；否則依不同prime個數奇偶給正負。排序不丟重數。",
        "en": "The empty prime multiset for1 gives mu1=1.A repeated prime gives a square divisor and value0;otherwise distinct-prime parity determines the sign.Sorting must not discard multiplicities."
      },
      "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": "S117",
      "uses": [
        "G001",
        "S116"
      ],
      "before": "Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor7 := div(462, 7)",
      "after": "Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor7 = 66",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor7，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor7 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "462",
          "7"
        ],
        "output": "66",
        "label": "Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor7"
      }
    },
    {
      "id": "S118",
      "uses": [
        "G001",
        "S117"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/7):actual positive input=66;check complete integer range1..66",
      "after": "ALL positive divisors INCREASING=[1, 2, 3, 6, 11, 22, 33, 66];not only primes/proper divisors or all integers<=n",
      "relation": "implies",
      "rule_id": "mobinv_ALL_positive_divisors_complete_indexing",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整範圍查整除，保1與n；d|n不是d<=n，也不只取prime或proper divisor。",
        "en": "Check divisibility over the complete range,retaining1 and n.The relation d|n is neither d<=n nor a prime/proper-divisor restriction."
      },
      "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": "S119",
      "uses": [
        "G001",
        "S118"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/7) f(1) ORIGINAL a times divisor := mul(20, 1)",
      "after": "Mobius inversion ORIGINAL inverse F(n/7) f(1) ORIGINAL a times divisor = 20",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/7) f(1) ORIGINAL a times divisor，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/7) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "20",
          "1"
        ],
        "output": "20",
        "label": "Mobius inversion ORIGINAL inverse F(n/7) f(1) ORIGINAL a times divisor"
      }
    },
    {
      "id": "S120",
      "uses": [
        "G001",
        "S119"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/7) f(1) plus ORIGINAL b := sum(20, 1)",
      "after": "Mobius inversion ORIGINAL inverse F(n/7) f(1) plus ORIGINAL b = 21",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/7) f(1) plus ORIGINAL b，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/7) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "20",
          "1"
        ],
        "output": "21",
        "label": "Mobius inversion ORIGINAL inverse F(n/7) f(1) plus ORIGINAL b"
      }
    },
    {
      "id": "S121",
      "uses": [
        "G001",
        "S120"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/7) f(2) ORIGINAL a times divisor := mul(20, 2)",
      "after": "Mobius inversion ORIGINAL inverse F(n/7) f(2) ORIGINAL a times divisor = 40",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/7) f(2) ORIGINAL a times divisor，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/7) f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "20",
          "2"
        ],
        "output": "40",
        "label": "Mobius inversion ORIGINAL inverse F(n/7) f(2) ORIGINAL a times divisor"
      }
    },
    {
      "id": "S122",
      "uses": [
        "G001",
        "S121"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/7) f(2) plus ORIGINAL b := sum(40, 1)",
      "after": "Mobius inversion ORIGINAL inverse F(n/7) f(2) plus ORIGINAL b = 41",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/7) f(2) plus ORIGINAL b，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/7) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "40",
          "1"
        ],
        "output": "41",
        "label": "Mobius inversion ORIGINAL inverse F(n/7) f(2) plus ORIGINAL b"
      }
    },
    {
      "id": "S123",
      "uses": [
        "G001",
        "S122"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/7) f(3) ORIGINAL a times divisor := mul(20, 3)",
      "after": "Mobius inversion ORIGINAL inverse F(n/7) f(3) ORIGINAL a times divisor = 60",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/7) f(3) ORIGINAL a times divisor，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/7) f(3) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "20",
          "3"
        ],
        "output": "60",
        "label": "Mobius inversion ORIGINAL inverse F(n/7) f(3) ORIGINAL a times divisor"
      }
    },
    {
      "id": "S124",
      "uses": [
        "G001",
        "S123"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/7) f(3) plus ORIGINAL b := sum(60, 1)",
      "after": "Mobius inversion ORIGINAL inverse F(n/7) f(3) plus ORIGINAL b = 61",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/7) f(3) plus ORIGINAL b，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/7) f(3) plus ORIGINAL b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "60",
          "1"
        ],
        "output": "61",
        "label": "Mobius inversion ORIGINAL inverse F(n/7) f(3) plus ORIGINAL b"
      }
    },
    {
      "id": "S125",
      "uses": [
        "G001",
        "S124"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/7) f(6) ORIGINAL a times divisor := mul(20, 6)",
      "after": "Mobius inversion ORIGINAL inverse F(n/7) f(6) ORIGINAL a times divisor = 120",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/7) f(6) ORIGINAL a times divisor，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/7) f(6) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "20",
          "6"
        ],
        "output": "120",
        "label": "Mobius inversion ORIGINAL inverse F(n/7) f(6) ORIGINAL a times divisor"
      }
    },
    {
      "id": "S126",
      "uses": [
        "G001",
        "S125"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/7) f(6) plus ORIGINAL b := sum(120, 1)",
      "after": "Mobius inversion ORIGINAL inverse F(n/7) f(6) plus ORIGINAL b = 121",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/7) f(6) plus ORIGINAL b，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/7) f(6) plus ORIGINAL b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "120",
          "1"
        ],
        "output": "121",
        "label": "Mobius inversion ORIGINAL inverse F(n/7) f(6) plus ORIGINAL b"
      }
    },
    {
      "id": "S127",
      "uses": [
        "G001",
        "S126"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/7) f(11) ORIGINAL a times divisor := mul(20, 11)",
      "after": "Mobius inversion ORIGINAL inverse F(n/7) f(11) ORIGINAL a times divisor = 220",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/7) f(11) ORIGINAL a times divisor，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/7) f(11) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "20",
          "11"
        ],
        "output": "220",
        "label": "Mobius inversion ORIGINAL inverse F(n/7) f(11) ORIGINAL a times divisor"
      }
    },
    {
      "id": "S128",
      "uses": [
        "G001",
        "S127"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/7) f(11) plus ORIGINAL b := sum(220, 1)",
      "after": "Mobius inversion ORIGINAL inverse F(n/7) f(11) plus ORIGINAL b = 221",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/7) f(11) plus ORIGINAL b，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/7) f(11) plus ORIGINAL b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "220",
          "1"
        ],
        "output": "221",
        "label": "Mobius inversion ORIGINAL inverse F(n/7) f(11) plus ORIGINAL b"
      }
    },
    {
      "id": "S129",
      "uses": [
        "G001",
        "S128"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/7) f(22) ORIGINAL a times divisor := mul(20, 22)",
      "after": "Mobius inversion ORIGINAL inverse F(n/7) f(22) ORIGINAL a times divisor = 440",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/7) f(22) ORIGINAL a times divisor，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/7) f(22) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "20",
          "22"
        ],
        "output": "440",
        "label": "Mobius inversion ORIGINAL inverse F(n/7) f(22) ORIGINAL a times divisor"
      }
    },
    {
      "id": "S130",
      "uses": [
        "G001",
        "S129"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/7) f(22) plus ORIGINAL b := sum(440, 1)",
      "after": "Mobius inversion ORIGINAL inverse F(n/7) f(22) plus ORIGINAL b = 441",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/7) f(22) plus ORIGINAL b，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/7) f(22) plus ORIGINAL b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "440",
          "1"
        ],
        "output": "441",
        "label": "Mobius inversion ORIGINAL inverse F(n/7) f(22) plus ORIGINAL b"
      }
    },
    {
      "id": "S131",
      "uses": [
        "G001",
        "S130"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/7) f(33) ORIGINAL a times divisor := mul(20, 33)",
      "after": "Mobius inversion ORIGINAL inverse F(n/7) f(33) ORIGINAL a times divisor = 660",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/7) f(33) ORIGINAL a times divisor，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/7) f(33) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "20",
          "33"
        ],
        "output": "660",
        "label": "Mobius inversion ORIGINAL inverse F(n/7) f(33) ORIGINAL a times divisor"
      }
    },
    {
      "id": "S132",
      "uses": [
        "G001",
        "S131"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/7) f(33) plus ORIGINAL b := sum(660, 1)",
      "after": "Mobius inversion ORIGINAL inverse F(n/7) f(33) plus ORIGINAL b = 661",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/7) f(33) plus ORIGINAL b，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/7) f(33) plus ORIGINAL b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "660",
          "1"
        ],
        "output": "661",
        "label": "Mobius inversion ORIGINAL inverse F(n/7) f(33) plus ORIGINAL b"
      }
    },
    {
      "id": "S133",
      "uses": [
        "G001",
        "S132"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/7) f(66) ORIGINAL a times divisor := mul(20, 66)",
      "after": "Mobius inversion ORIGINAL inverse F(n/7) f(66) ORIGINAL a times divisor = 1320",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/7) f(66) ORIGINAL a times divisor，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/7) f(66) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "20",
          "66"
        ],
        "output": "1320",
        "label": "Mobius inversion ORIGINAL inverse F(n/7) f(66) ORIGINAL a times divisor"
      }
    },
    {
      "id": "S134",
      "uses": [
        "G001",
        "S133"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/7) f(66) plus ORIGINAL b := sum(1320, 1)",
      "after": "Mobius inversion ORIGINAL inverse F(n/7) f(66) plus ORIGINAL b = 1321",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/7) f(66) plus ORIGINAL b，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/7) f(66) plus ORIGINAL b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1320",
          "1"
        ],
        "output": "1321",
        "label": "Mobius inversion ORIGINAL inverse F(n/7) f(66) plus ORIGINAL b"
      }
    },
    {
      "id": "S135",
      "uses": [
        "G001",
        "S134"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/7) full forward divisor sum := sum(21, 41, 61, 121, 221, 441, 661, 1321)",
      "after": "Mobius inversion ORIGINAL inverse F(n/7) full forward divisor sum = 2888",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/7) full forward divisor sum，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/7) full forward divisor sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "21",
          "41",
          "61",
          "121",
          "221",
          "441",
          "661",
          "1321"
        ],
        "output": "2888",
        "label": "Mobius inversion ORIGINAL inverse F(n/7) full forward divisor sum"
      }
    },
    {
      "id": "S136",
      "uses": [
        "G001",
        "S135"
      ],
      "before": "Mobius inversion ORIGINAL inverse mu(7) times SAME F(n/7) := mul(-1, 2888)",
      "after": "Mobius inversion ORIGINAL inverse mu(7) times SAME F(n/7) = -2888",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse mu(7) times SAME F(n/7)，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse mu(7) times SAME F(n/7) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-1",
          "2888"
        ],
        "output": "-2888",
        "label": "Mobius inversion ORIGINAL inverse mu(7) times SAME F(n/7)"
      }
    },
    {
      "id": "S137",
      "uses": [
        "G001",
        "S136"
      ],
      "before": "Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor11:actual positive input for mu=11",
      "after": "complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search",
      "relation": "implies",
      "rule_id": "mobius_complete_UPF_input_role",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先明示正在算原n還是其正因數的mu；每次需要完整prime重數，不能用找到的幾個因數猜squarefree。",
        "en": "Identify whether the input is the original n or one of its positive divisors.Every mu evaluation requires complete prime multiplicities,not a partial factor search."
      },
      "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": "S138",
      "uses": [
        "G001",
        "S137"
      ],
      "before": "ORIGINAL positive integer absolute magnitude=11>1",
      "after": "initialize remaining=n,prime multiset empty;invariant product(extracted)*remaining=n;every smaller trial divisor has been exhausted",
      "relation": "implies",
      "rule_id": "upf_factorization_remaining_product_invariant",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每次只除真正整除的d，並以已窮盡更小因數維持d為prime的理由，不先假定原n為prime。",
        "en": "Divide only by an actual divisor and exhaust smaller candidates,which justifies primality of each extracted divisor without assuming n is prime."
      },
      "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": "S139",
      "uses": [
        "G001",
        "S138"
      ],
      "before": "next divisor=2;previous remaining=11;all smaller divisors exhausted",
      "after": "extracted multiplicity=0;next remaining=11;remaining modulo2=1;extracted multiset=[]",
      "relation": "implies",
      "rule_id": "upf_complete_trial_division_remaining_invariant",
      "conditions": [],
      "explanation": {
        "zh-Hant": "除盡同一prime的所有重數才前進；不把n的原平方根與remaining的停止責任混淆，也不漏最後的prime remainder。",
        "en": "Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted."
      },
      "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": "S140",
      "uses": [
        "G001",
        "S139"
      ],
      "before": "next divisor=3;previous remaining=11;all smaller divisors exhausted",
      "after": "extracted multiplicity=0;next remaining=11;remaining modulo3=2;extracted multiset=[]",
      "relation": "implies",
      "rule_id": "upf_complete_trial_division_remaining_invariant",
      "conditions": [],
      "explanation": {
        "zh-Hant": "除盡同一prime的所有重數才前進；不把n的原平方根與remaining的停止責任混淆，也不漏最後的prime remainder。",
        "en": "Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted."
      },
      "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": "S141",
      "uses": [
        "G001",
        "S140"
      ],
      "before": "upf canonical complete product factor0 := mul(1, 11)",
      "after": "upf canonical complete product factor0 = 11",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。",
        "en": "Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 upf canonical complete product factor0，不以小數近似替代。",
          "en": "Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "11"
        ],
        "output": "11",
        "label": "upf canonical complete product factor0"
      }
    },
    {
      "id": "S142",
      "uses": [
        "G001",
        "S141"
      ],
      "before": "exhausted trial divisors below4;terminal remaining=11;d*d>remaining;ORIGINAL n=11",
      "after": "complete increasing prime multiset=[11];full product=11;a composite terminal remainder would have an already tested smaller divisor",
      "relation": "implies",
      "rule_id": "upf_final_prime_remainder_complete_existence_instance",
      "conditions": [],
      "explanation": {
        "zh-Hant": "若remaining>1仍合數，其最小prime因數會在已驗範圍，與除盡不變式矛盾，所以補入的末項為prime。這證本例完整分解；一般存在也可對n作強歸納，合數拆成較小正因數後遞歸。",
        "en": "If a terminal remainder greater than1 were composite,its least prime factor would already have been exhausted,a contradiction.The final entry is therefore prime.This certifies this instance;general existence follows by strong induction,recursively splitting a composite into smaller positive factors."
      },
      "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": "S143",
      "uses": [
        "G001",
        "S142"
      ],
      "before": "Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor11:input=11;COMPLETE prime multiset=[11];distinct primes=[11]",
      "after": "mu=-1;empty product case=否;repeated prime=否",
      "relation": "implies",
      "rule_id": "mobius_value_one_repeated_prime_or_squarefree_parity",
      "conditions": [],
      "explanation": {
        "zh-Hant": "1的空prime多重集給mu1=1；任何prime重複表示square divisor而給0；否則依不同prime個數奇偶給正負。排序不丟重數。",
        "en": "The empty prime multiset for1 gives mu1=1.A repeated prime gives a square divisor and value0;otherwise distinct-prime parity determines the sign.Sorting must not discard multiplicities."
      },
      "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": "S144",
      "uses": [
        "G001",
        "S143"
      ],
      "before": "Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor11 := div(462, 11)",
      "after": "Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor11 = 42",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor11，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor11 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "462",
          "11"
        ],
        "output": "42",
        "label": "Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor11"
      }
    },
    {
      "id": "S145",
      "uses": [
        "G001",
        "S144"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/11):actual positive input=42;check complete integer range1..42",
      "after": "ALL positive divisors INCREASING=[1, 2, 3, 6, 7, 14, 21, 42];not only primes/proper divisors or all integers<=n",
      "relation": "implies",
      "rule_id": "mobinv_ALL_positive_divisors_complete_indexing",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整範圍查整除，保1與n；d|n不是d<=n，也不只取prime或proper divisor。",
        "en": "Check divisibility over the complete range,retaining1 and n.The relation d|n is neither d<=n nor a prime/proper-divisor restriction."
      },
      "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": "S146",
      "uses": [
        "G001",
        "S145"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/11) f(1) ORIGINAL a times divisor := mul(20, 1)",
      "after": "Mobius inversion ORIGINAL inverse F(n/11) f(1) ORIGINAL a times divisor = 20",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/11) f(1) ORIGINAL a times divisor，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/11) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "20",
          "1"
        ],
        "output": "20",
        "label": "Mobius inversion ORIGINAL inverse F(n/11) f(1) ORIGINAL a times divisor"
      }
    },
    {
      "id": "S147",
      "uses": [
        "G001",
        "S146"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/11) f(1) plus ORIGINAL b := sum(20, 1)",
      "after": "Mobius inversion ORIGINAL inverse F(n/11) f(1) plus ORIGINAL b = 21",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/11) f(1) plus ORIGINAL b，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/11) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "20",
          "1"
        ],
        "output": "21",
        "label": "Mobius inversion ORIGINAL inverse F(n/11) f(1) plus ORIGINAL b"
      }
    },
    {
      "id": "S148",
      "uses": [
        "G001",
        "S147"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/11) f(2) ORIGINAL a times divisor := mul(20, 2)",
      "after": "Mobius inversion ORIGINAL inverse F(n/11) f(2) ORIGINAL a times divisor = 40",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/11) f(2) ORIGINAL a times divisor，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/11) f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "20",
          "2"
        ],
        "output": "40",
        "label": "Mobius inversion ORIGINAL inverse F(n/11) f(2) ORIGINAL a times divisor"
      }
    },
    {
      "id": "S149",
      "uses": [
        "G001",
        "S148"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/11) f(2) plus ORIGINAL b := sum(40, 1)",
      "after": "Mobius inversion ORIGINAL inverse F(n/11) f(2) plus ORIGINAL b = 41",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/11) f(2) plus ORIGINAL b，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/11) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "40",
          "1"
        ],
        "output": "41",
        "label": "Mobius inversion ORIGINAL inverse F(n/11) f(2) plus ORIGINAL b"
      }
    },
    {
      "id": "S150",
      "uses": [
        "G001",
        "S149"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/11) f(3) ORIGINAL a times divisor := mul(20, 3)",
      "after": "Mobius inversion ORIGINAL inverse F(n/11) f(3) ORIGINAL a times divisor = 60",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/11) f(3) ORIGINAL a times divisor，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/11) f(3) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "20",
          "3"
        ],
        "output": "60",
        "label": "Mobius inversion ORIGINAL inverse F(n/11) f(3) ORIGINAL a times divisor"
      }
    },
    {
      "id": "S151",
      "uses": [
        "G001",
        "S150"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/11) f(3) plus ORIGINAL b := sum(60, 1)",
      "after": "Mobius inversion ORIGINAL inverse F(n/11) f(3) plus ORIGINAL b = 61",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/11) f(3) plus ORIGINAL b，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/11) f(3) plus ORIGINAL b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "60",
          "1"
        ],
        "output": "61",
        "label": "Mobius inversion ORIGINAL inverse F(n/11) f(3) plus ORIGINAL b"
      }
    },
    {
      "id": "S152",
      "uses": [
        "G001",
        "S151"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/11) f(6) ORIGINAL a times divisor := mul(20, 6)",
      "after": "Mobius inversion ORIGINAL inverse F(n/11) f(6) ORIGINAL a times divisor = 120",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/11) f(6) ORIGINAL a times divisor，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/11) f(6) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "20",
          "6"
        ],
        "output": "120",
        "label": "Mobius inversion ORIGINAL inverse F(n/11) f(6) ORIGINAL a times divisor"
      }
    },
    {
      "id": "S153",
      "uses": [
        "G001",
        "S152"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/11) f(6) plus ORIGINAL b := sum(120, 1)",
      "after": "Mobius inversion ORIGINAL inverse F(n/11) f(6) plus ORIGINAL b = 121",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/11) f(6) plus ORIGINAL b，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/11) f(6) plus ORIGINAL b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "120",
          "1"
        ],
        "output": "121",
        "label": "Mobius inversion ORIGINAL inverse F(n/11) f(6) plus ORIGINAL b"
      }
    },
    {
      "id": "S154",
      "uses": [
        "G001",
        "S153"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/11) f(7) ORIGINAL a times divisor := mul(20, 7)",
      "after": "Mobius inversion ORIGINAL inverse F(n/11) f(7) ORIGINAL a times divisor = 140",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/11) f(7) ORIGINAL a times divisor，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/11) f(7) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "20",
          "7"
        ],
        "output": "140",
        "label": "Mobius inversion ORIGINAL inverse F(n/11) f(7) ORIGINAL a times divisor"
      }
    },
    {
      "id": "S155",
      "uses": [
        "G001",
        "S154"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/11) f(7) plus ORIGINAL b := sum(140, 1)",
      "after": "Mobius inversion ORIGINAL inverse F(n/11) f(7) plus ORIGINAL b = 141",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/11) f(7) plus ORIGINAL b，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/11) f(7) plus ORIGINAL b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "140",
          "1"
        ],
        "output": "141",
        "label": "Mobius inversion ORIGINAL inverse F(n/11) f(7) plus ORIGINAL b"
      }
    },
    {
      "id": "S156",
      "uses": [
        "G001",
        "S155"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/11) f(14) ORIGINAL a times divisor := mul(20, 14)",
      "after": "Mobius inversion ORIGINAL inverse F(n/11) f(14) ORIGINAL a times divisor = 280",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/11) f(14) ORIGINAL a times divisor，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/11) f(14) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "20",
          "14"
        ],
        "output": "280",
        "label": "Mobius inversion ORIGINAL inverse F(n/11) f(14) ORIGINAL a times divisor"
      }
    },
    {
      "id": "S157",
      "uses": [
        "G001",
        "S156"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/11) f(14) plus ORIGINAL b := sum(280, 1)",
      "after": "Mobius inversion ORIGINAL inverse F(n/11) f(14) plus ORIGINAL b = 281",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/11) f(14) plus ORIGINAL b，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/11) f(14) plus ORIGINAL b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "280",
          "1"
        ],
        "output": "281",
        "label": "Mobius inversion ORIGINAL inverse F(n/11) f(14) plus ORIGINAL b"
      }
    },
    {
      "id": "S158",
      "uses": [
        "G001",
        "S157"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/11) f(21) ORIGINAL a times divisor := mul(20, 21)",
      "after": "Mobius inversion ORIGINAL inverse F(n/11) f(21) ORIGINAL a times divisor = 420",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/11) f(21) ORIGINAL a times divisor，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/11) f(21) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "20",
          "21"
        ],
        "output": "420",
        "label": "Mobius inversion ORIGINAL inverse F(n/11) f(21) ORIGINAL a times divisor"
      }
    },
    {
      "id": "S159",
      "uses": [
        "G001",
        "S158"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/11) f(21) plus ORIGINAL b := sum(420, 1)",
      "after": "Mobius inversion ORIGINAL inverse F(n/11) f(21) plus ORIGINAL b = 421",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/11) f(21) plus ORIGINAL b，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/11) f(21) plus ORIGINAL b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "420",
          "1"
        ],
        "output": "421",
        "label": "Mobius inversion ORIGINAL inverse F(n/11) f(21) plus ORIGINAL b"
      }
    },
    {
      "id": "S160",
      "uses": [
        "G001",
        "S159"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/11) f(42) ORIGINAL a times divisor := mul(20, 42)",
      "after": "Mobius inversion ORIGINAL inverse F(n/11) f(42) ORIGINAL a times divisor = 840",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/11) f(42) ORIGINAL a times divisor，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/11) f(42) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "20",
          "42"
        ],
        "output": "840",
        "label": "Mobius inversion ORIGINAL inverse F(n/11) f(42) ORIGINAL a times divisor"
      }
    },
    {
      "id": "S161",
      "uses": [
        "G001",
        "S160"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/11) f(42) plus ORIGINAL b := sum(840, 1)",
      "after": "Mobius inversion ORIGINAL inverse F(n/11) f(42) plus ORIGINAL b = 841",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/11) f(42) plus ORIGINAL b，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/11) f(42) plus ORIGINAL b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "840",
          "1"
        ],
        "output": "841",
        "label": "Mobius inversion ORIGINAL inverse F(n/11) f(42) plus ORIGINAL b"
      }
    },
    {
      "id": "S162",
      "uses": [
        "G001",
        "S161"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/11) full forward divisor sum := sum(21, 41, 61, 121, 141, 281, 421, 841)",
      "after": "Mobius inversion ORIGINAL inverse F(n/11) full forward divisor sum = 1928",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/11) full forward divisor sum，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/11) full forward divisor sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "21",
          "41",
          "61",
          "121",
          "141",
          "281",
          "421",
          "841"
        ],
        "output": "1928",
        "label": "Mobius inversion ORIGINAL inverse F(n/11) full forward divisor sum"
      }
    },
    {
      "id": "S163",
      "uses": [
        "G001",
        "S162"
      ],
      "before": "Mobius inversion ORIGINAL inverse mu(11) times SAME F(n/11) := mul(-1, 1928)",
      "after": "Mobius inversion ORIGINAL inverse mu(11) times SAME F(n/11) = -1928",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse mu(11) times SAME F(n/11)，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse mu(11) times SAME F(n/11) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-1",
          "1928"
        ],
        "output": "-1928",
        "label": "Mobius inversion ORIGINAL inverse mu(11) times SAME F(n/11)"
      }
    },
    {
      "id": "S164",
      "uses": [
        "G001",
        "S163"
      ],
      "before": "Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor14:actual positive input for mu=14",
      "after": "complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search",
      "relation": "implies",
      "rule_id": "mobius_complete_UPF_input_role",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先明示正在算原n還是其正因數的mu；每次需要完整prime重數，不能用找到的幾個因數猜squarefree。",
        "en": "Identify whether the input is the original n or one of its positive divisors.Every mu evaluation requires complete prime multiplicities,not a partial factor search."
      },
      "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": "S165",
      "uses": [
        "G001",
        "S164"
      ],
      "before": "ORIGINAL positive integer absolute magnitude=14>1",
      "after": "initialize remaining=n,prime multiset empty;invariant product(extracted)*remaining=n;every smaller trial divisor has been exhausted",
      "relation": "implies",
      "rule_id": "upf_factorization_remaining_product_invariant",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每次只除真正整除的d，並以已窮盡更小因數維持d為prime的理由，不先假定原n為prime。",
        "en": "Divide only by an actual divisor and exhaust smaller candidates,which justifies primality of each extracted divisor without assuming n is prime."
      },
      "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": "S166",
      "uses": [
        "G001",
        "S165"
      ],
      "before": "upf extract divisor2 occurrence0 := div(14, 2)",
      "after": "upf extract divisor2 occurrence0 = 7",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。",
        "en": "Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 upf extract divisor2 occurrence0，不以小數近似替代。",
          "en": "Recompute upf extract divisor2 occurrence0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "14",
          "2"
        ],
        "output": "7",
        "label": "upf extract divisor2 occurrence0"
      }
    },
    {
      "id": "S167",
      "uses": [
        "G001",
        "S166"
      ],
      "before": "next divisor=2;previous remaining=14;all smaller divisors exhausted",
      "after": "extracted multiplicity=1;next remaining=7;remaining modulo2=1;extracted multiset=[2]",
      "relation": "implies",
      "rule_id": "upf_complete_trial_division_remaining_invariant",
      "conditions": [],
      "explanation": {
        "zh-Hant": "除盡同一prime的所有重數才前進；不把n的原平方根與remaining的停止責任混淆，也不漏最後的prime remainder。",
        "en": "Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted."
      },
      "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": "S168",
      "uses": [
        "G001",
        "S167"
      ],
      "before": "upf canonical complete product factor0 := mul(1, 2)",
      "after": "upf canonical complete product factor0 = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。",
        "en": "Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 upf canonical complete product factor0，不以小數近似替代。",
          "en": "Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "2"
        ],
        "output": "2",
        "label": "upf canonical complete product factor0"
      }
    },
    {
      "id": "S169",
      "uses": [
        "G001",
        "S168"
      ],
      "before": "upf canonical complete product factor1 := mul(2, 7)",
      "after": "upf canonical complete product factor1 = 14",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。",
        "en": "Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 upf canonical complete product factor1，不以小數近似替代。",
          "en": "Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "7"
        ],
        "output": "14",
        "label": "upf canonical complete product factor1"
      }
    },
    {
      "id": "S170",
      "uses": [
        "G001",
        "S169"
      ],
      "before": "exhausted trial divisors below3;terminal remaining=7;d*d>remaining;ORIGINAL n=14",
      "after": "complete increasing prime multiset=[2, 7];full product=14;a composite terminal remainder would have an already tested smaller divisor",
      "relation": "implies",
      "rule_id": "upf_final_prime_remainder_complete_existence_instance",
      "conditions": [],
      "explanation": {
        "zh-Hant": "若remaining>1仍合數，其最小prime因數會在已驗範圍，與除盡不變式矛盾，所以補入的末項為prime。這證本例完整分解；一般存在也可對n作強歸納，合數拆成較小正因數後遞歸。",
        "en": "If a terminal remainder greater than1 were composite,its least prime factor would already have been exhausted,a contradiction.The final entry is therefore prime.This certifies this instance;general existence follows by strong induction,recursively splitting a composite into smaller positive factors."
      },
      "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": "S171",
      "uses": [
        "G001",
        "S170"
      ],
      "before": "Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor14:input=14;COMPLETE prime multiset=[2, 7];distinct primes=[2, 7]",
      "after": "mu=1;empty product case=否;repeated prime=否",
      "relation": "implies",
      "rule_id": "mobius_value_one_repeated_prime_or_squarefree_parity",
      "conditions": [],
      "explanation": {
        "zh-Hant": "1的空prime多重集給mu1=1；任何prime重複表示square divisor而給0；否則依不同prime個數奇偶給正負。排序不丟重數。",
        "en": "The empty prime multiset for1 gives mu1=1.A repeated prime gives a square divisor and value0;otherwise distinct-prime parity determines the sign.Sorting must not discard multiplicities."
      },
      "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": "S172",
      "uses": [
        "G001",
        "S171"
      ],
      "before": "Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor14 := div(462, 14)",
      "after": "Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor14 = 33",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor14，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor14 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "462",
          "14"
        ],
        "output": "33",
        "label": "Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor14"
      }
    },
    {
      "id": "S173",
      "uses": [
        "G001",
        "S172"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/14):actual positive input=33;check complete integer range1..33",
      "after": "ALL positive divisors INCREASING=[1, 3, 11, 33];not only primes/proper divisors or all integers<=n",
      "relation": "implies",
      "rule_id": "mobinv_ALL_positive_divisors_complete_indexing",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整範圍查整除，保1與n；d|n不是d<=n，也不只取prime或proper divisor。",
        "en": "Check divisibility over the complete range,retaining1 and n.The relation d|n is neither d<=n nor a prime/proper-divisor restriction."
      },
      "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": "S174",
      "uses": [
        "G001",
        "S173"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/14) f(1) ORIGINAL a times divisor := mul(20, 1)",
      "after": "Mobius inversion ORIGINAL inverse F(n/14) f(1) ORIGINAL a times divisor = 20",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/14) f(1) ORIGINAL a times divisor，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/14) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "20",
          "1"
        ],
        "output": "20",
        "label": "Mobius inversion ORIGINAL inverse F(n/14) f(1) ORIGINAL a times divisor"
      }
    },
    {
      "id": "S175",
      "uses": [
        "G001",
        "S174"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/14) f(1) plus ORIGINAL b := sum(20, 1)",
      "after": "Mobius inversion ORIGINAL inverse F(n/14) f(1) plus ORIGINAL b = 21",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/14) f(1) plus ORIGINAL b，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/14) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "20",
          "1"
        ],
        "output": "21",
        "label": "Mobius inversion ORIGINAL inverse F(n/14) f(1) plus ORIGINAL b"
      }
    },
    {
      "id": "S176",
      "uses": [
        "G001",
        "S175"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/14) f(3) ORIGINAL a times divisor := mul(20, 3)",
      "after": "Mobius inversion ORIGINAL inverse F(n/14) f(3) ORIGINAL a times divisor = 60",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/14) f(3) ORIGINAL a times divisor，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/14) f(3) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "20",
          "3"
        ],
        "output": "60",
        "label": "Mobius inversion ORIGINAL inverse F(n/14) f(3) ORIGINAL a times divisor"
      }
    },
    {
      "id": "S177",
      "uses": [
        "G001",
        "S176"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/14) f(3) plus ORIGINAL b := sum(60, 1)",
      "after": "Mobius inversion ORIGINAL inverse F(n/14) f(3) plus ORIGINAL b = 61",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/14) f(3) plus ORIGINAL b，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/14) f(3) plus ORIGINAL b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "60",
          "1"
        ],
        "output": "61",
        "label": "Mobius inversion ORIGINAL inverse F(n/14) f(3) plus ORIGINAL b"
      }
    },
    {
      "id": "S178",
      "uses": [
        "G001",
        "S177"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/14) f(11) ORIGINAL a times divisor := mul(20, 11)",
      "after": "Mobius inversion ORIGINAL inverse F(n/14) f(11) ORIGINAL a times divisor = 220",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/14) f(11) ORIGINAL a times divisor，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/14) f(11) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "20",
          "11"
        ],
        "output": "220",
        "label": "Mobius inversion ORIGINAL inverse F(n/14) f(11) ORIGINAL a times divisor"
      }
    },
    {
      "id": "S179",
      "uses": [
        "G001",
        "S178"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/14) f(11) plus ORIGINAL b := sum(220, 1)",
      "after": "Mobius inversion ORIGINAL inverse F(n/14) f(11) plus ORIGINAL b = 221",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/14) f(11) plus ORIGINAL b，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/14) f(11) plus ORIGINAL b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "220",
          "1"
        ],
        "output": "221",
        "label": "Mobius inversion ORIGINAL inverse F(n/14) f(11) plus ORIGINAL b"
      }
    },
    {
      "id": "S180",
      "uses": [
        "G001",
        "S179"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/14) f(33) ORIGINAL a times divisor := mul(20, 33)",
      "after": "Mobius inversion ORIGINAL inverse F(n/14) f(33) ORIGINAL a times divisor = 660",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/14) f(33) ORIGINAL a times divisor，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/14) f(33) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "20",
          "33"
        ],
        "output": "660",
        "label": "Mobius inversion ORIGINAL inverse F(n/14) f(33) ORIGINAL a times divisor"
      }
    },
    {
      "id": "S181",
      "uses": [
        "G001",
        "S180"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/14) f(33) plus ORIGINAL b := sum(660, 1)",
      "after": "Mobius inversion ORIGINAL inverse F(n/14) f(33) plus ORIGINAL b = 661",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/14) f(33) plus ORIGINAL b，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/14) f(33) plus ORIGINAL b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "660",
          "1"
        ],
        "output": "661",
        "label": "Mobius inversion ORIGINAL inverse F(n/14) f(33) plus ORIGINAL b"
      }
    },
    {
      "id": "S182",
      "uses": [
        "G001",
        "S181"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/14) full forward divisor sum := sum(21, 61, 221, 661)",
      "after": "Mobius inversion ORIGINAL inverse F(n/14) full forward divisor sum = 964",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/14) full forward divisor sum，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/14) full forward divisor sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "21",
          "61",
          "221",
          "661"
        ],
        "output": "964",
        "label": "Mobius inversion ORIGINAL inverse F(n/14) full forward divisor sum"
      }
    },
    {
      "id": "S183",
      "uses": [
        "G001",
        "S182"
      ],
      "before": "Mobius inversion ORIGINAL inverse mu(14) times SAME F(n/14) := mul(1, 964)",
      "after": "Mobius inversion ORIGINAL inverse mu(14) times SAME F(n/14) = 964",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse mu(14) times SAME F(n/14)，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse mu(14) times SAME F(n/14) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "964"
        ],
        "output": "964",
        "label": "Mobius inversion ORIGINAL inverse mu(14) times SAME F(n/14)"
      }
    },
    {
      "id": "S184",
      "uses": [
        "G001",
        "S183"
      ],
      "before": "Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor21:actual positive input for mu=21",
      "after": "complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search",
      "relation": "implies",
      "rule_id": "mobius_complete_UPF_input_role",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先明示正在算原n還是其正因數的mu；每次需要完整prime重數，不能用找到的幾個因數猜squarefree。",
        "en": "Identify whether the input is the original n or one of its positive divisors.Every mu evaluation requires complete prime multiplicities,not a partial factor search."
      },
      "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": "S185",
      "uses": [
        "G001",
        "S184"
      ],
      "before": "ORIGINAL positive integer absolute magnitude=21>1",
      "after": "initialize remaining=n,prime multiset empty;invariant product(extracted)*remaining=n;every smaller trial divisor has been exhausted",
      "relation": "implies",
      "rule_id": "upf_factorization_remaining_product_invariant",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每次只除真正整除的d，並以已窮盡更小因數維持d為prime的理由，不先假定原n為prime。",
        "en": "Divide only by an actual divisor and exhaust smaller candidates,which justifies primality of each extracted divisor without assuming n is prime."
      },
      "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": "S186",
      "uses": [
        "G001",
        "S185"
      ],
      "before": "next divisor=2;previous remaining=21;all smaller divisors exhausted",
      "after": "extracted multiplicity=0;next remaining=21;remaining modulo2=1;extracted multiset=[]",
      "relation": "implies",
      "rule_id": "upf_complete_trial_division_remaining_invariant",
      "conditions": [],
      "explanation": {
        "zh-Hant": "除盡同一prime的所有重數才前進；不把n的原平方根與remaining的停止責任混淆，也不漏最後的prime remainder。",
        "en": "Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted."
      },
      "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": "S187",
      "uses": [
        "G001",
        "S186"
      ],
      "before": "upf extract divisor3 occurrence0 := div(21, 3)",
      "after": "upf extract divisor3 occurrence0 = 7",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。",
        "en": "Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 upf extract divisor3 occurrence0，不以小數近似替代。",
          "en": "Recompute upf extract divisor3 occurrence0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "21",
          "3"
        ],
        "output": "7",
        "label": "upf extract divisor3 occurrence0"
      }
    },
    {
      "id": "S188",
      "uses": [
        "G001",
        "S187"
      ],
      "before": "next divisor=3;previous remaining=21;all smaller divisors exhausted",
      "after": "extracted multiplicity=1;next remaining=7;remaining modulo3=1;extracted multiset=[3]",
      "relation": "implies",
      "rule_id": "upf_complete_trial_division_remaining_invariant",
      "conditions": [],
      "explanation": {
        "zh-Hant": "除盡同一prime的所有重數才前進；不把n的原平方根與remaining的停止責任混淆，也不漏最後的prime remainder。",
        "en": "Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted."
      },
      "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": "S189",
      "uses": [
        "G001",
        "S188"
      ],
      "before": "upf canonical complete product factor0 := mul(1, 3)",
      "after": "upf canonical complete product factor0 = 3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。",
        "en": "Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 upf canonical complete product factor0，不以小數近似替代。",
          "en": "Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "3"
        ],
        "output": "3",
        "label": "upf canonical complete product factor0"
      }
    },
    {
      "id": "S190",
      "uses": [
        "G001",
        "S189"
      ],
      "before": "upf canonical complete product factor1 := mul(3, 7)",
      "after": "upf canonical complete product factor1 = 21",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。",
        "en": "Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 upf canonical complete product factor1，不以小數近似替代。",
          "en": "Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "3",
          "7"
        ],
        "output": "21",
        "label": "upf canonical complete product factor1"
      }
    },
    {
      "id": "S191",
      "uses": [
        "G001",
        "S190"
      ],
      "before": "exhausted trial divisors below4;terminal remaining=7;d*d>remaining;ORIGINAL n=21",
      "after": "complete increasing prime multiset=[3, 7];full product=21;a composite terminal remainder would have an already tested smaller divisor",
      "relation": "implies",
      "rule_id": "upf_final_prime_remainder_complete_existence_instance",
      "conditions": [],
      "explanation": {
        "zh-Hant": "若remaining>1仍合數，其最小prime因數會在已驗範圍，與除盡不變式矛盾，所以補入的末項為prime。這證本例完整分解；一般存在也可對n作強歸納，合數拆成較小正因數後遞歸。",
        "en": "If a terminal remainder greater than1 were composite,its least prime factor would already have been exhausted,a contradiction.The final entry is therefore prime.This certifies this instance;general existence follows by strong induction,recursively splitting a composite into smaller positive factors."
      },
      "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": "S192",
      "uses": [
        "G001",
        "S191"
      ],
      "before": "Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor21:input=21;COMPLETE prime multiset=[3, 7];distinct primes=[3, 7]",
      "after": "mu=1;empty product case=否;repeated prime=否",
      "relation": "implies",
      "rule_id": "mobius_value_one_repeated_prime_or_squarefree_parity",
      "conditions": [],
      "explanation": {
        "zh-Hant": "1的空prime多重集給mu1=1；任何prime重複表示square divisor而給0；否則依不同prime個數奇偶給正負。排序不丟重數。",
        "en": "The empty prime multiset for1 gives mu1=1.A repeated prime gives a square divisor and value0;otherwise distinct-prime parity determines the sign.Sorting must not discard multiplicities."
      },
      "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": "S193",
      "uses": [
        "G001",
        "S192"
      ],
      "before": "Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor21 := div(462, 21)",
      "after": "Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor21 = 22",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor21，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor21 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "462",
          "21"
        ],
        "output": "22",
        "label": "Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor21"
      }
    },
    {
      "id": "S194",
      "uses": [
        "G001",
        "S193"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/21):actual positive input=22;check complete integer range1..22",
      "after": "ALL positive divisors INCREASING=[1, 2, 11, 22];not only primes/proper divisors or all integers<=n",
      "relation": "implies",
      "rule_id": "mobinv_ALL_positive_divisors_complete_indexing",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整範圍查整除，保1與n；d|n不是d<=n，也不只取prime或proper divisor。",
        "en": "Check divisibility over the complete range,retaining1 and n.The relation d|n is neither d<=n nor a prime/proper-divisor restriction."
      },
      "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": "S195",
      "uses": [
        "G001",
        "S194"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/21) f(1) ORIGINAL a times divisor := mul(20, 1)",
      "after": "Mobius inversion ORIGINAL inverse F(n/21) f(1) ORIGINAL a times divisor = 20",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/21) f(1) ORIGINAL a times divisor，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/21) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "20",
          "1"
        ],
        "output": "20",
        "label": "Mobius inversion ORIGINAL inverse F(n/21) f(1) ORIGINAL a times divisor"
      }
    },
    {
      "id": "S196",
      "uses": [
        "G001",
        "S195"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/21) f(1) plus ORIGINAL b := sum(20, 1)",
      "after": "Mobius inversion ORIGINAL inverse F(n/21) f(1) plus ORIGINAL b = 21",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/21) f(1) plus ORIGINAL b，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/21) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "20",
          "1"
        ],
        "output": "21",
        "label": "Mobius inversion ORIGINAL inverse F(n/21) f(1) plus ORIGINAL b"
      }
    },
    {
      "id": "S197",
      "uses": [
        "G001",
        "S196"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/21) f(2) ORIGINAL a times divisor := mul(20, 2)",
      "after": "Mobius inversion ORIGINAL inverse F(n/21) f(2) ORIGINAL a times divisor = 40",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/21) f(2) ORIGINAL a times divisor，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/21) f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "20",
          "2"
        ],
        "output": "40",
        "label": "Mobius inversion ORIGINAL inverse F(n/21) f(2) ORIGINAL a times divisor"
      }
    },
    {
      "id": "S198",
      "uses": [
        "G001",
        "S197"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/21) f(2) plus ORIGINAL b := sum(40, 1)",
      "after": "Mobius inversion ORIGINAL inverse F(n/21) f(2) plus ORIGINAL b = 41",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/21) f(2) plus ORIGINAL b，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/21) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "40",
          "1"
        ],
        "output": "41",
        "label": "Mobius inversion ORIGINAL inverse F(n/21) f(2) plus ORIGINAL b"
      }
    },
    {
      "id": "S199",
      "uses": [
        "G001",
        "S198"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/21) f(11) ORIGINAL a times divisor := mul(20, 11)",
      "after": "Mobius inversion ORIGINAL inverse F(n/21) f(11) ORIGINAL a times divisor = 220",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/21) f(11) ORIGINAL a times divisor，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/21) f(11) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "20",
          "11"
        ],
        "output": "220",
        "label": "Mobius inversion ORIGINAL inverse F(n/21) f(11) ORIGINAL a times divisor"
      }
    },
    {
      "id": "S200",
      "uses": [
        "G001",
        "S199"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/21) f(11) plus ORIGINAL b := sum(220, 1)",
      "after": "Mobius inversion ORIGINAL inverse F(n/21) f(11) plus ORIGINAL b = 221",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/21) f(11) plus ORIGINAL b，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/21) f(11) plus ORIGINAL b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "220",
          "1"
        ],
        "output": "221",
        "label": "Mobius inversion ORIGINAL inverse F(n/21) f(11) plus ORIGINAL b"
      }
    },
    {
      "id": "S201",
      "uses": [
        "G001",
        "S200"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/21) f(22) ORIGINAL a times divisor := mul(20, 22)",
      "after": "Mobius inversion ORIGINAL inverse F(n/21) f(22) ORIGINAL a times divisor = 440",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/21) f(22) ORIGINAL a times divisor，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/21) f(22) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "20",
          "22"
        ],
        "output": "440",
        "label": "Mobius inversion ORIGINAL inverse F(n/21) f(22) ORIGINAL a times divisor"
      }
    },
    {
      "id": "S202",
      "uses": [
        "G001",
        "S201"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/21) f(22) plus ORIGINAL b := sum(440, 1)",
      "after": "Mobius inversion ORIGINAL inverse F(n/21) f(22) plus ORIGINAL b = 441",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/21) f(22) plus ORIGINAL b，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/21) f(22) plus ORIGINAL b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "440",
          "1"
        ],
        "output": "441",
        "label": "Mobius inversion ORIGINAL inverse F(n/21) f(22) plus ORIGINAL b"
      }
    },
    {
      "id": "S203",
      "uses": [
        "G001",
        "S202"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/21) full forward divisor sum := sum(21, 41, 221, 441)",
      "after": "Mobius inversion ORIGINAL inverse F(n/21) full forward divisor sum = 724",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/21) full forward divisor sum，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/21) full forward divisor sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "21",
          "41",
          "221",
          "441"
        ],
        "output": "724",
        "label": "Mobius inversion ORIGINAL inverse F(n/21) full forward divisor sum"
      }
    },
    {
      "id": "S204",
      "uses": [
        "G001",
        "S203"
      ],
      "before": "Mobius inversion ORIGINAL inverse mu(21) times SAME F(n/21) := mul(1, 724)",
      "after": "Mobius inversion ORIGINAL inverse mu(21) times SAME F(n/21) = 724",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse mu(21) times SAME F(n/21)，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse mu(21) times SAME F(n/21) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "724"
        ],
        "output": "724",
        "label": "Mobius inversion ORIGINAL inverse mu(21) times SAME F(n/21)"
      }
    },
    {
      "id": "S205",
      "uses": [
        "G001",
        "S204"
      ],
      "before": "Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor22:actual positive input for mu=22",
      "after": "complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search",
      "relation": "implies",
      "rule_id": "mobius_complete_UPF_input_role",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先明示正在算原n還是其正因數的mu；每次需要完整prime重數，不能用找到的幾個因數猜squarefree。",
        "en": "Identify whether the input is the original n or one of its positive divisors.Every mu evaluation requires complete prime multiplicities,not a partial factor search."
      },
      "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": "S206",
      "uses": [
        "G001",
        "S205"
      ],
      "before": "ORIGINAL positive integer absolute magnitude=22>1",
      "after": "initialize remaining=n,prime multiset empty;invariant product(extracted)*remaining=n;every smaller trial divisor has been exhausted",
      "relation": "implies",
      "rule_id": "upf_factorization_remaining_product_invariant",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每次只除真正整除的d，並以已窮盡更小因數維持d為prime的理由，不先假定原n為prime。",
        "en": "Divide only by an actual divisor and exhaust smaller candidates,which justifies primality of each extracted divisor without assuming n is prime."
      },
      "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": "S207",
      "uses": [
        "G001",
        "S206"
      ],
      "before": "upf extract divisor2 occurrence0 := div(22, 2)",
      "after": "upf extract divisor2 occurrence0 = 11",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。",
        "en": "Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 upf extract divisor2 occurrence0，不以小數近似替代。",
          "en": "Recompute upf extract divisor2 occurrence0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "22",
          "2"
        ],
        "output": "11",
        "label": "upf extract divisor2 occurrence0"
      }
    },
    {
      "id": "S208",
      "uses": [
        "G001",
        "S207"
      ],
      "before": "next divisor=2;previous remaining=22;all smaller divisors exhausted",
      "after": "extracted multiplicity=1;next remaining=11;remaining modulo2=1;extracted multiset=[2]",
      "relation": "implies",
      "rule_id": "upf_complete_trial_division_remaining_invariant",
      "conditions": [],
      "explanation": {
        "zh-Hant": "除盡同一prime的所有重數才前進；不把n的原平方根與remaining的停止責任混淆，也不漏最後的prime remainder。",
        "en": "Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted."
      },
      "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": "S209",
      "uses": [
        "G001",
        "S208"
      ],
      "before": "next divisor=3;previous remaining=11;all smaller divisors exhausted",
      "after": "extracted multiplicity=0;next remaining=11;remaining modulo3=2;extracted multiset=[2]",
      "relation": "implies",
      "rule_id": "upf_complete_trial_division_remaining_invariant",
      "conditions": [],
      "explanation": {
        "zh-Hant": "除盡同一prime的所有重數才前進；不把n的原平方根與remaining的停止責任混淆，也不漏最後的prime remainder。",
        "en": "Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted."
      },
      "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": "S210",
      "uses": [
        "G001",
        "S209"
      ],
      "before": "upf canonical complete product factor0 := mul(1, 2)",
      "after": "upf canonical complete product factor0 = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。",
        "en": "Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 upf canonical complete product factor0，不以小數近似替代。",
          "en": "Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "2"
        ],
        "output": "2",
        "label": "upf canonical complete product factor0"
      }
    },
    {
      "id": "S211",
      "uses": [
        "G001",
        "S210"
      ],
      "before": "upf canonical complete product factor1 := mul(2, 11)",
      "after": "upf canonical complete product factor1 = 22",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。",
        "en": "Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 upf canonical complete product factor1，不以小數近似替代。",
          "en": "Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "11"
        ],
        "output": "22",
        "label": "upf canonical complete product factor1"
      }
    },
    {
      "id": "S212",
      "uses": [
        "G001",
        "S211"
      ],
      "before": "exhausted trial divisors below4;terminal remaining=11;d*d>remaining;ORIGINAL n=22",
      "after": "complete increasing prime multiset=[2, 11];full product=22;a composite terminal remainder would have an already tested smaller divisor",
      "relation": "implies",
      "rule_id": "upf_final_prime_remainder_complete_existence_instance",
      "conditions": [],
      "explanation": {
        "zh-Hant": "若remaining>1仍合數，其最小prime因數會在已驗範圍，與除盡不變式矛盾，所以補入的末項為prime。這證本例完整分解；一般存在也可對n作強歸納，合數拆成較小正因數後遞歸。",
        "en": "If a terminal remainder greater than1 were composite,its least prime factor would already have been exhausted,a contradiction.The final entry is therefore prime.This certifies this instance;general existence follows by strong induction,recursively splitting a composite into smaller positive factors."
      },
      "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": "S213",
      "uses": [
        "G001",
        "S212"
      ],
      "before": "Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor22:input=22;COMPLETE prime multiset=[2, 11];distinct primes=[2, 11]",
      "after": "mu=1;empty product case=否;repeated prime=否",
      "relation": "implies",
      "rule_id": "mobius_value_one_repeated_prime_or_squarefree_parity",
      "conditions": [],
      "explanation": {
        "zh-Hant": "1的空prime多重集給mu1=1；任何prime重複表示square divisor而給0；否則依不同prime個數奇偶給正負。排序不丟重數。",
        "en": "The empty prime multiset for1 gives mu1=1.A repeated prime gives a square divisor and value0;otherwise distinct-prime parity determines the sign.Sorting must not discard multiplicities."
      },
      "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": "S214",
      "uses": [
        "G001",
        "S213"
      ],
      "before": "Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor22 := div(462, 22)",
      "after": "Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor22 = 21",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor22，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor22 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "462",
          "22"
        ],
        "output": "21",
        "label": "Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor22"
      }
    },
    {
      "id": "S215",
      "uses": [
        "G001",
        "S214"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/22):actual positive input=21;check complete integer range1..21",
      "after": "ALL positive divisors INCREASING=[1, 3, 7, 21];not only primes/proper divisors or all integers<=n",
      "relation": "implies",
      "rule_id": "mobinv_ALL_positive_divisors_complete_indexing",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整範圍查整除，保1與n；d|n不是d<=n，也不只取prime或proper divisor。",
        "en": "Check divisibility over the complete range,retaining1 and n.The relation d|n is neither d<=n nor a prime/proper-divisor restriction."
      },
      "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": "S216",
      "uses": [
        "G001",
        "S215"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/22) f(1) ORIGINAL a times divisor := mul(20, 1)",
      "after": "Mobius inversion ORIGINAL inverse F(n/22) f(1) ORIGINAL a times divisor = 20",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/22) f(1) ORIGINAL a times divisor，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/22) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "20",
          "1"
        ],
        "output": "20",
        "label": "Mobius inversion ORIGINAL inverse F(n/22) f(1) ORIGINAL a times divisor"
      }
    },
    {
      "id": "S217",
      "uses": [
        "G001",
        "S216"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/22) f(1) plus ORIGINAL b := sum(20, 1)",
      "after": "Mobius inversion ORIGINAL inverse F(n/22) f(1) plus ORIGINAL b = 21",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/22) f(1) plus ORIGINAL b，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/22) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "20",
          "1"
        ],
        "output": "21",
        "label": "Mobius inversion ORIGINAL inverse F(n/22) f(1) plus ORIGINAL b"
      }
    },
    {
      "id": "S218",
      "uses": [
        "G001",
        "S217"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/22) f(3) ORIGINAL a times divisor := mul(20, 3)",
      "after": "Mobius inversion ORIGINAL inverse F(n/22) f(3) ORIGINAL a times divisor = 60",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/22) f(3) ORIGINAL a times divisor，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/22) f(3) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "20",
          "3"
        ],
        "output": "60",
        "label": "Mobius inversion ORIGINAL inverse F(n/22) f(3) ORIGINAL a times divisor"
      }
    },
    {
      "id": "S219",
      "uses": [
        "G001",
        "S218"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/22) f(3) plus ORIGINAL b := sum(60, 1)",
      "after": "Mobius inversion ORIGINAL inverse F(n/22) f(3) plus ORIGINAL b = 61",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/22) f(3) plus ORIGINAL b，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/22) f(3) plus ORIGINAL b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "60",
          "1"
        ],
        "output": "61",
        "label": "Mobius inversion ORIGINAL inverse F(n/22) f(3) plus ORIGINAL b"
      }
    },
    {
      "id": "S220",
      "uses": [
        "G001",
        "S219"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/22) f(7) ORIGINAL a times divisor := mul(20, 7)",
      "after": "Mobius inversion ORIGINAL inverse F(n/22) f(7) ORIGINAL a times divisor = 140",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/22) f(7) ORIGINAL a times divisor，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/22) f(7) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "20",
          "7"
        ],
        "output": "140",
        "label": "Mobius inversion ORIGINAL inverse F(n/22) f(7) ORIGINAL a times divisor"
      }
    },
    {
      "id": "S221",
      "uses": [
        "G001",
        "S220"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/22) f(7) plus ORIGINAL b := sum(140, 1)",
      "after": "Mobius inversion ORIGINAL inverse F(n/22) f(7) plus ORIGINAL b = 141",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/22) f(7) plus ORIGINAL b，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/22) f(7) plus ORIGINAL b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "140",
          "1"
        ],
        "output": "141",
        "label": "Mobius inversion ORIGINAL inverse F(n/22) f(7) plus ORIGINAL b"
      }
    },
    {
      "id": "S222",
      "uses": [
        "G001",
        "S221"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/22) f(21) ORIGINAL a times divisor := mul(20, 21)",
      "after": "Mobius inversion ORIGINAL inverse F(n/22) f(21) ORIGINAL a times divisor = 420",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/22) f(21) ORIGINAL a times divisor，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/22) f(21) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "20",
          "21"
        ],
        "output": "420",
        "label": "Mobius inversion ORIGINAL inverse F(n/22) f(21) ORIGINAL a times divisor"
      }
    },
    {
      "id": "S223",
      "uses": [
        "G001",
        "S222"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/22) f(21) plus ORIGINAL b := sum(420, 1)",
      "after": "Mobius inversion ORIGINAL inverse F(n/22) f(21) plus ORIGINAL b = 421",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/22) f(21) plus ORIGINAL b，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/22) f(21) plus ORIGINAL b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "420",
          "1"
        ],
        "output": "421",
        "label": "Mobius inversion ORIGINAL inverse F(n/22) f(21) plus ORIGINAL b"
      }
    },
    {
      "id": "S224",
      "uses": [
        "G001",
        "S223"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/22) full forward divisor sum := sum(21, 61, 141, 421)",
      "after": "Mobius inversion ORIGINAL inverse F(n/22) full forward divisor sum = 644",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/22) full forward divisor sum，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/22) full forward divisor sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "21",
          "61",
          "141",
          "421"
        ],
        "output": "644",
        "label": "Mobius inversion ORIGINAL inverse F(n/22) full forward divisor sum"
      }
    },
    {
      "id": "S225",
      "uses": [
        "G001",
        "S224"
      ],
      "before": "Mobius inversion ORIGINAL inverse mu(22) times SAME F(n/22) := mul(1, 644)",
      "after": "Mobius inversion ORIGINAL inverse mu(22) times SAME F(n/22) = 644",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse mu(22) times SAME F(n/22)，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse mu(22) times SAME F(n/22) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "644"
        ],
        "output": "644",
        "label": "Mobius inversion ORIGINAL inverse mu(22) times SAME F(n/22)"
      }
    },
    {
      "id": "S226",
      "uses": [
        "G001",
        "S225"
      ],
      "before": "Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor33:actual positive input for mu=33",
      "after": "complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search",
      "relation": "implies",
      "rule_id": "mobius_complete_UPF_input_role",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先明示正在算原n還是其正因數的mu；每次需要完整prime重數，不能用找到的幾個因數猜squarefree。",
        "en": "Identify whether the input is the original n or one of its positive divisors.Every mu evaluation requires complete prime multiplicities,not a partial factor search."
      },
      "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": "S227",
      "uses": [
        "G001",
        "S226"
      ],
      "before": "ORIGINAL positive integer absolute magnitude=33>1",
      "after": "initialize remaining=n,prime multiset empty;invariant product(extracted)*remaining=n;every smaller trial divisor has been exhausted",
      "relation": "implies",
      "rule_id": "upf_factorization_remaining_product_invariant",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每次只除真正整除的d，並以已窮盡更小因數維持d為prime的理由，不先假定原n為prime。",
        "en": "Divide only by an actual divisor and exhaust smaller candidates,which justifies primality of each extracted divisor without assuming n is prime."
      },
      "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": "S228",
      "uses": [
        "G001",
        "S227"
      ],
      "before": "next divisor=2;previous remaining=33;all smaller divisors exhausted",
      "after": "extracted multiplicity=0;next remaining=33;remaining modulo2=1;extracted multiset=[]",
      "relation": "implies",
      "rule_id": "upf_complete_trial_division_remaining_invariant",
      "conditions": [],
      "explanation": {
        "zh-Hant": "除盡同一prime的所有重數才前進；不把n的原平方根與remaining的停止責任混淆，也不漏最後的prime remainder。",
        "en": "Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted."
      },
      "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": "S229",
      "uses": [
        "G001",
        "S228"
      ],
      "before": "upf extract divisor3 occurrence0 := div(33, 3)",
      "after": "upf extract divisor3 occurrence0 = 11",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。",
        "en": "Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 upf extract divisor3 occurrence0，不以小數近似替代。",
          "en": "Recompute upf extract divisor3 occurrence0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "33",
          "3"
        ],
        "output": "11",
        "label": "upf extract divisor3 occurrence0"
      }
    },
    {
      "id": "S230",
      "uses": [
        "G001",
        "S229"
      ],
      "before": "next divisor=3;previous remaining=33;all smaller divisors exhausted",
      "after": "extracted multiplicity=1;next remaining=11;remaining modulo3=2;extracted multiset=[3]",
      "relation": "implies",
      "rule_id": "upf_complete_trial_division_remaining_invariant",
      "conditions": [],
      "explanation": {
        "zh-Hant": "除盡同一prime的所有重數才前進；不把n的原平方根與remaining的停止責任混淆，也不漏最後的prime remainder。",
        "en": "Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted."
      },
      "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": "S231",
      "uses": [
        "G001",
        "S230"
      ],
      "before": "upf canonical complete product factor0 := mul(1, 3)",
      "after": "upf canonical complete product factor0 = 3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。",
        "en": "Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 upf canonical complete product factor0，不以小數近似替代。",
          "en": "Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "3"
        ],
        "output": "3",
        "label": "upf canonical complete product factor0"
      }
    },
    {
      "id": "S232",
      "uses": [
        "G001",
        "S231"
      ],
      "before": "upf canonical complete product factor1 := mul(3, 11)",
      "after": "upf canonical complete product factor1 = 33",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。",
        "en": "Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 upf canonical complete product factor1，不以小數近似替代。",
          "en": "Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "3",
          "11"
        ],
        "output": "33",
        "label": "upf canonical complete product factor1"
      }
    },
    {
      "id": "S233",
      "uses": [
        "G001",
        "S232"
      ],
      "before": "exhausted trial divisors below4;terminal remaining=11;d*d>remaining;ORIGINAL n=33",
      "after": "complete increasing prime multiset=[3, 11];full product=33;a composite terminal remainder would have an already tested smaller divisor",
      "relation": "implies",
      "rule_id": "upf_final_prime_remainder_complete_existence_instance",
      "conditions": [],
      "explanation": {
        "zh-Hant": "若remaining>1仍合數，其最小prime因數會在已驗範圍，與除盡不變式矛盾，所以補入的末項為prime。這證本例完整分解；一般存在也可對n作強歸納，合數拆成較小正因數後遞歸。",
        "en": "If a terminal remainder greater than1 were composite,its least prime factor would already have been exhausted,a contradiction.The final entry is therefore prime.This certifies this instance;general existence follows by strong induction,recursively splitting a composite into smaller positive factors."
      },
      "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": "S234",
      "uses": [
        "G001",
        "S233"
      ],
      "before": "Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor33:input=33;COMPLETE prime multiset=[3, 11];distinct primes=[3, 11]",
      "after": "mu=1;empty product case=否;repeated prime=否",
      "relation": "implies",
      "rule_id": "mobius_value_one_repeated_prime_or_squarefree_parity",
      "conditions": [],
      "explanation": {
        "zh-Hant": "1的空prime多重集給mu1=1；任何prime重複表示square divisor而給0；否則依不同prime個數奇偶給正負。排序不丟重數。",
        "en": "The empty prime multiset for1 gives mu1=1.A repeated prime gives a square divisor and value0;otherwise distinct-prime parity determines the sign.Sorting must not discard multiplicities."
      },
      "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": "S235",
      "uses": [
        "G001",
        "S234"
      ],
      "before": "Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor33 := div(462, 33)",
      "after": "Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor33 = 14",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor33，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor33 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "462",
          "33"
        ],
        "output": "14",
        "label": "Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor33"
      }
    },
    {
      "id": "S236",
      "uses": [
        "G001",
        "S235"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/33):actual positive input=14;check complete integer range1..14",
      "after": "ALL positive divisors INCREASING=[1, 2, 7, 14];not only primes/proper divisors or all integers<=n",
      "relation": "implies",
      "rule_id": "mobinv_ALL_positive_divisors_complete_indexing",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整範圍查整除，保1與n；d|n不是d<=n，也不只取prime或proper divisor。",
        "en": "Check divisibility over the complete range,retaining1 and n.The relation d|n is neither d<=n nor a prime/proper-divisor restriction."
      },
      "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": "S237",
      "uses": [
        "G001",
        "S236"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/33) f(1) ORIGINAL a times divisor := mul(20, 1)",
      "after": "Mobius inversion ORIGINAL inverse F(n/33) f(1) ORIGINAL a times divisor = 20",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/33) f(1) ORIGINAL a times divisor，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/33) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "20",
          "1"
        ],
        "output": "20",
        "label": "Mobius inversion ORIGINAL inverse F(n/33) f(1) ORIGINAL a times divisor"
      }
    },
    {
      "id": "S238",
      "uses": [
        "G001",
        "S237"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/33) f(1) plus ORIGINAL b := sum(20, 1)",
      "after": "Mobius inversion ORIGINAL inverse F(n/33) f(1) plus ORIGINAL b = 21",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/33) f(1) plus ORIGINAL b，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/33) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "20",
          "1"
        ],
        "output": "21",
        "label": "Mobius inversion ORIGINAL inverse F(n/33) f(1) plus ORIGINAL b"
      }
    },
    {
      "id": "S239",
      "uses": [
        "G001",
        "S238"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/33) f(2) ORIGINAL a times divisor := mul(20, 2)",
      "after": "Mobius inversion ORIGINAL inverse F(n/33) f(2) ORIGINAL a times divisor = 40",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/33) f(2) ORIGINAL a times divisor，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/33) f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "20",
          "2"
        ],
        "output": "40",
        "label": "Mobius inversion ORIGINAL inverse F(n/33) f(2) ORIGINAL a times divisor"
      }
    },
    {
      "id": "S240",
      "uses": [
        "G001",
        "S239"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/33) f(2) plus ORIGINAL b := sum(40, 1)",
      "after": "Mobius inversion ORIGINAL inverse F(n/33) f(2) plus ORIGINAL b = 41",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/33) f(2) plus ORIGINAL b，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/33) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "40",
          "1"
        ],
        "output": "41",
        "label": "Mobius inversion ORIGINAL inverse F(n/33) f(2) plus ORIGINAL b"
      }
    },
    {
      "id": "S241",
      "uses": [
        "G001",
        "S240"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/33) f(7) ORIGINAL a times divisor := mul(20, 7)",
      "after": "Mobius inversion ORIGINAL inverse F(n/33) f(7) ORIGINAL a times divisor = 140",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/33) f(7) ORIGINAL a times divisor，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/33) f(7) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "20",
          "7"
        ],
        "output": "140",
        "label": "Mobius inversion ORIGINAL inverse F(n/33) f(7) ORIGINAL a times divisor"
      }
    },
    {
      "id": "S242",
      "uses": [
        "G001",
        "S241"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/33) f(7) plus ORIGINAL b := sum(140, 1)",
      "after": "Mobius inversion ORIGINAL inverse F(n/33) f(7) plus ORIGINAL b = 141",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/33) f(7) plus ORIGINAL b，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/33) f(7) plus ORIGINAL b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "140",
          "1"
        ],
        "output": "141",
        "label": "Mobius inversion ORIGINAL inverse F(n/33) f(7) plus ORIGINAL b"
      }
    },
    {
      "id": "S243",
      "uses": [
        "G001",
        "S242"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/33) f(14) ORIGINAL a times divisor := mul(20, 14)",
      "after": "Mobius inversion ORIGINAL inverse F(n/33) f(14) ORIGINAL a times divisor = 280",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/33) f(14) ORIGINAL a times divisor，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/33) f(14) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "20",
          "14"
        ],
        "output": "280",
        "label": "Mobius inversion ORIGINAL inverse F(n/33) f(14) ORIGINAL a times divisor"
      }
    },
    {
      "id": "S244",
      "uses": [
        "G001",
        "S243"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/33) f(14) plus ORIGINAL b := sum(280, 1)",
      "after": "Mobius inversion ORIGINAL inverse F(n/33) f(14) plus ORIGINAL b = 281",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/33) f(14) plus ORIGINAL b，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/33) f(14) plus ORIGINAL b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "280",
          "1"
        ],
        "output": "281",
        "label": "Mobius inversion ORIGINAL inverse F(n/33) f(14) plus ORIGINAL b"
      }
    },
    {
      "id": "S245",
      "uses": [
        "G001",
        "S244"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/33) full forward divisor sum := sum(21, 41, 141, 281)",
      "after": "Mobius inversion ORIGINAL inverse F(n/33) full forward divisor sum = 484",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/33) full forward divisor sum，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/33) full forward divisor sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "21",
          "41",
          "141",
          "281"
        ],
        "output": "484",
        "label": "Mobius inversion ORIGINAL inverse F(n/33) full forward divisor sum"
      }
    },
    {
      "id": "S246",
      "uses": [
        "G001",
        "S245"
      ],
      "before": "Mobius inversion ORIGINAL inverse mu(33) times SAME F(n/33) := mul(1, 484)",
      "after": "Mobius inversion ORIGINAL inverse mu(33) times SAME F(n/33) = 484",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse mu(33) times SAME F(n/33)，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse mu(33) times SAME F(n/33) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "484"
        ],
        "output": "484",
        "label": "Mobius inversion ORIGINAL inverse mu(33) times SAME F(n/33)"
      }
    },
    {
      "id": "S247",
      "uses": [
        "G001",
        "S246"
      ],
      "before": "Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor42:actual positive input for mu=42",
      "after": "complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search",
      "relation": "implies",
      "rule_id": "mobius_complete_UPF_input_role",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先明示正在算原n還是其正因數的mu；每次需要完整prime重數，不能用找到的幾個因數猜squarefree。",
        "en": "Identify whether the input is the original n or one of its positive divisors.Every mu evaluation requires complete prime multiplicities,not a partial factor search."
      },
      "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": "S248",
      "uses": [
        "G001",
        "S247"
      ],
      "before": "ORIGINAL positive integer absolute magnitude=42>1",
      "after": "initialize remaining=n,prime multiset empty;invariant product(extracted)*remaining=n;every smaller trial divisor has been exhausted",
      "relation": "implies",
      "rule_id": "upf_factorization_remaining_product_invariant",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每次只除真正整除的d，並以已窮盡更小因數維持d為prime的理由，不先假定原n為prime。",
        "en": "Divide only by an actual divisor and exhaust smaller candidates,which justifies primality of each extracted divisor without assuming n is prime."
      },
      "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": "S249",
      "uses": [
        "G001",
        "S248"
      ],
      "before": "upf extract divisor2 occurrence0 := div(42, 2)",
      "after": "upf extract divisor2 occurrence0 = 21",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。",
        "en": "Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 upf extract divisor2 occurrence0，不以小數近似替代。",
          "en": "Recompute upf extract divisor2 occurrence0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "42",
          "2"
        ],
        "output": "21",
        "label": "upf extract divisor2 occurrence0"
      }
    },
    {
      "id": "S250",
      "uses": [
        "G001",
        "S249"
      ],
      "before": "next divisor=2;previous remaining=42;all smaller divisors exhausted",
      "after": "extracted multiplicity=1;next remaining=21;remaining modulo2=1;extracted multiset=[2]",
      "relation": "implies",
      "rule_id": "upf_complete_trial_division_remaining_invariant",
      "conditions": [],
      "explanation": {
        "zh-Hant": "除盡同一prime的所有重數才前進；不把n的原平方根與remaining的停止責任混淆，也不漏最後的prime remainder。",
        "en": "Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted."
      },
      "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": "S251",
      "uses": [
        "G001",
        "S250"
      ],
      "before": "upf extract divisor3 occurrence1 := div(21, 3)",
      "after": "upf extract divisor3 occurrence1 = 7",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。",
        "en": "Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 upf extract divisor3 occurrence1，不以小數近似替代。",
          "en": "Recompute upf extract divisor3 occurrence1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "21",
          "3"
        ],
        "output": "7",
        "label": "upf extract divisor3 occurrence1"
      }
    },
    {
      "id": "S252",
      "uses": [
        "G001",
        "S251"
      ],
      "before": "next divisor=3;previous remaining=21;all smaller divisors exhausted",
      "after": "extracted multiplicity=1;next remaining=7;remaining modulo3=1;extracted multiset=[2, 3]",
      "relation": "implies",
      "rule_id": "upf_complete_trial_division_remaining_invariant",
      "conditions": [],
      "explanation": {
        "zh-Hant": "除盡同一prime的所有重數才前進；不把n的原平方根與remaining的停止責任混淆，也不漏最後的prime remainder。",
        "en": "Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted."
      },
      "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": "S253",
      "uses": [
        "G001",
        "S252"
      ],
      "before": "upf canonical complete product factor0 := mul(1, 2)",
      "after": "upf canonical complete product factor0 = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。",
        "en": "Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 upf canonical complete product factor0，不以小數近似替代。",
          "en": "Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "2"
        ],
        "output": "2",
        "label": "upf canonical complete product factor0"
      }
    },
    {
      "id": "S254",
      "uses": [
        "G001",
        "S253"
      ],
      "before": "upf canonical complete product factor1 := mul(2, 3)",
      "after": "upf canonical complete product factor1 = 6",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。",
        "en": "Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 upf canonical complete product factor1，不以小數近似替代。",
          "en": "Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "3"
        ],
        "output": "6",
        "label": "upf canonical complete product factor1"
      }
    },
    {
      "id": "S255",
      "uses": [
        "G001",
        "S254"
      ],
      "before": "upf canonical complete product factor2 := mul(6, 7)",
      "after": "upf canonical complete product factor2 = 42",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。",
        "en": "Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 upf canonical complete product factor2，不以小數近似替代。",
          "en": "Recompute upf canonical complete product factor2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "6",
          "7"
        ],
        "output": "42",
        "label": "upf canonical complete product factor2"
      }
    },
    {
      "id": "S256",
      "uses": [
        "G001",
        "S255"
      ],
      "before": "exhausted trial divisors below4;terminal remaining=7;d*d>remaining;ORIGINAL n=42",
      "after": "complete increasing prime multiset=[2, 3, 7];full product=42;a composite terminal remainder would have an already tested smaller divisor",
      "relation": "implies",
      "rule_id": "upf_final_prime_remainder_complete_existence_instance",
      "conditions": [],
      "explanation": {
        "zh-Hant": "若remaining>1仍合數，其最小prime因數會在已驗範圍，與除盡不變式矛盾，所以補入的末項為prime。這證本例完整分解；一般存在也可對n作強歸納，合數拆成較小正因數後遞歸。",
        "en": "If a terminal remainder greater than1 were composite,its least prime factor would already have been exhausted,a contradiction.The final entry is therefore prime.This certifies this instance;general existence follows by strong induction,recursively splitting a composite into smaller positive factors."
      },
      "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": "S257",
      "uses": [
        "G001",
        "S256"
      ],
      "before": "Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor42:input=42;COMPLETE prime multiset=[2, 3, 7];distinct primes=[2, 3, 7]",
      "after": "mu=-1;empty product case=否;repeated prime=否",
      "relation": "implies",
      "rule_id": "mobius_value_one_repeated_prime_or_squarefree_parity",
      "conditions": [],
      "explanation": {
        "zh-Hant": "1的空prime多重集給mu1=1；任何prime重複表示square divisor而給0；否則依不同prime個數奇偶給正負。排序不丟重數。",
        "en": "The empty prime multiset for1 gives mu1=1.A repeated prime gives a square divisor and value0;otherwise distinct-prime parity determines the sign.Sorting must not discard multiplicities."
      },
      "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": "S258",
      "uses": [
        "G001",
        "S257"
      ],
      "before": "Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor42 := div(462, 42)",
      "after": "Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor42 = 11",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor42，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor42 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "462",
          "42"
        ],
        "output": "11",
        "label": "Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor42"
      }
    },
    {
      "id": "S259",
      "uses": [
        "G001",
        "S258"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/42):actual positive input=11;check complete integer range1..11",
      "after": "ALL positive divisors INCREASING=[1, 11];not only primes/proper divisors or all integers<=n",
      "relation": "implies",
      "rule_id": "mobinv_ALL_positive_divisors_complete_indexing",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整範圍查整除，保1與n；d|n不是d<=n，也不只取prime或proper divisor。",
        "en": "Check divisibility over the complete range,retaining1 and n.The relation d|n is neither d<=n nor a prime/proper-divisor restriction."
      },
      "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": "S260",
      "uses": [
        "G001",
        "S259"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/42) f(1) ORIGINAL a times divisor := mul(20, 1)",
      "after": "Mobius inversion ORIGINAL inverse F(n/42) f(1) ORIGINAL a times divisor = 20",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/42) f(1) ORIGINAL a times divisor，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/42) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "20",
          "1"
        ],
        "output": "20",
        "label": "Mobius inversion ORIGINAL inverse F(n/42) f(1) ORIGINAL a times divisor"
      }
    },
    {
      "id": "S261",
      "uses": [
        "G001",
        "S260"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/42) f(1) plus ORIGINAL b := sum(20, 1)",
      "after": "Mobius inversion ORIGINAL inverse F(n/42) f(1) plus ORIGINAL b = 21",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/42) f(1) plus ORIGINAL b，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/42) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "20",
          "1"
        ],
        "output": "21",
        "label": "Mobius inversion ORIGINAL inverse F(n/42) f(1) plus ORIGINAL b"
      }
    },
    {
      "id": "S262",
      "uses": [
        "G001",
        "S261"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/42) f(11) ORIGINAL a times divisor := mul(20, 11)",
      "after": "Mobius inversion ORIGINAL inverse F(n/42) f(11) ORIGINAL a times divisor = 220",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/42) f(11) ORIGINAL a times divisor，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/42) f(11) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "20",
          "11"
        ],
        "output": "220",
        "label": "Mobius inversion ORIGINAL inverse F(n/42) f(11) ORIGINAL a times divisor"
      }
    },
    {
      "id": "S263",
      "uses": [
        "G001",
        "S262"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/42) f(11) plus ORIGINAL b := sum(220, 1)",
      "after": "Mobius inversion ORIGINAL inverse F(n/42) f(11) plus ORIGINAL b = 221",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/42) f(11) plus ORIGINAL b，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/42) f(11) plus ORIGINAL b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "220",
          "1"
        ],
        "output": "221",
        "label": "Mobius inversion ORIGINAL inverse F(n/42) f(11) plus ORIGINAL b"
      }
    },
    {
      "id": "S264",
      "uses": [
        "G001",
        "S263"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/42) full forward divisor sum := sum(21, 221)",
      "after": "Mobius inversion ORIGINAL inverse F(n/42) full forward divisor sum = 242",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/42) full forward divisor sum，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/42) full forward divisor sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "21",
          "221"
        ],
        "output": "242",
        "label": "Mobius inversion ORIGINAL inverse F(n/42) full forward divisor sum"
      }
    },
    {
      "id": "S265",
      "uses": [
        "G001",
        "S264"
      ],
      "before": "Mobius inversion ORIGINAL inverse mu(42) times SAME F(n/42) := mul(-1, 242)",
      "after": "Mobius inversion ORIGINAL inverse mu(42) times SAME F(n/42) = -242",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse mu(42) times SAME F(n/42)，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse mu(42) times SAME F(n/42) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-1",
          "242"
        ],
        "output": "-242",
        "label": "Mobius inversion ORIGINAL inverse mu(42) times SAME F(n/42)"
      }
    },
    {
      "id": "S266",
      "uses": [
        "G001",
        "S265"
      ],
      "before": "Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor66:actual positive input for mu=66",
      "after": "complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search",
      "relation": "implies",
      "rule_id": "mobius_complete_UPF_input_role",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先明示正在算原n還是其正因數的mu；每次需要完整prime重數，不能用找到的幾個因數猜squarefree。",
        "en": "Identify whether the input is the original n or one of its positive divisors.Every mu evaluation requires complete prime multiplicities,not a partial factor search."
      },
      "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": "S267",
      "uses": [
        "G001",
        "S266"
      ],
      "before": "ORIGINAL positive integer absolute magnitude=66>1",
      "after": "initialize remaining=n,prime multiset empty;invariant product(extracted)*remaining=n;every smaller trial divisor has been exhausted",
      "relation": "implies",
      "rule_id": "upf_factorization_remaining_product_invariant",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每次只除真正整除的d，並以已窮盡更小因數維持d為prime的理由，不先假定原n為prime。",
        "en": "Divide only by an actual divisor and exhaust smaller candidates,which justifies primality of each extracted divisor without assuming n is prime."
      },
      "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": "S268",
      "uses": [
        "G001",
        "S267"
      ],
      "before": "upf extract divisor2 occurrence0 := div(66, 2)",
      "after": "upf extract divisor2 occurrence0 = 33",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。",
        "en": "Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 upf extract divisor2 occurrence0，不以小數近似替代。",
          "en": "Recompute upf extract divisor2 occurrence0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "66",
          "2"
        ],
        "output": "33",
        "label": "upf extract divisor2 occurrence0"
      }
    },
    {
      "id": "S269",
      "uses": [
        "G001",
        "S268"
      ],
      "before": "next divisor=2;previous remaining=66;all smaller divisors exhausted",
      "after": "extracted multiplicity=1;next remaining=33;remaining modulo2=1;extracted multiset=[2]",
      "relation": "implies",
      "rule_id": "upf_complete_trial_division_remaining_invariant",
      "conditions": [],
      "explanation": {
        "zh-Hant": "除盡同一prime的所有重數才前進；不把n的原平方根與remaining的停止責任混淆，也不漏最後的prime remainder。",
        "en": "Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted."
      },
      "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": "S270",
      "uses": [
        "G001",
        "S269"
      ],
      "before": "upf extract divisor3 occurrence1 := div(33, 3)",
      "after": "upf extract divisor3 occurrence1 = 11",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。",
        "en": "Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 upf extract divisor3 occurrence1，不以小數近似替代。",
          "en": "Recompute upf extract divisor3 occurrence1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "33",
          "3"
        ],
        "output": "11",
        "label": "upf extract divisor3 occurrence1"
      }
    },
    {
      "id": "S271",
      "uses": [
        "G001",
        "S270"
      ],
      "before": "next divisor=3;previous remaining=33;all smaller divisors exhausted",
      "after": "extracted multiplicity=1;next remaining=11;remaining modulo3=2;extracted multiset=[2, 3]",
      "relation": "implies",
      "rule_id": "upf_complete_trial_division_remaining_invariant",
      "conditions": [],
      "explanation": {
        "zh-Hant": "除盡同一prime的所有重數才前進；不把n的原平方根與remaining的停止責任混淆，也不漏最後的prime remainder。",
        "en": "Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted."
      },
      "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": "S272",
      "uses": [
        "G001",
        "S271"
      ],
      "before": "upf canonical complete product factor0 := mul(1, 2)",
      "after": "upf canonical complete product factor0 = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。",
        "en": "Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 upf canonical complete product factor0，不以小數近似替代。",
          "en": "Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "2"
        ],
        "output": "2",
        "label": "upf canonical complete product factor0"
      }
    },
    {
      "id": "S273",
      "uses": [
        "G001",
        "S272"
      ],
      "before": "upf canonical complete product factor1 := mul(2, 3)",
      "after": "upf canonical complete product factor1 = 6",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。",
        "en": "Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 upf canonical complete product factor1，不以小數近似替代。",
          "en": "Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "3"
        ],
        "output": "6",
        "label": "upf canonical complete product factor1"
      }
    },
    {
      "id": "S274",
      "uses": [
        "G001",
        "S273"
      ],
      "before": "upf canonical complete product factor2 := mul(6, 11)",
      "after": "upf canonical complete product factor2 = 66",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。",
        "en": "Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 upf canonical complete product factor2，不以小數近似替代。",
          "en": "Recompute upf canonical complete product factor2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "6",
          "11"
        ],
        "output": "66",
        "label": "upf canonical complete product factor2"
      }
    },
    {
      "id": "S275",
      "uses": [
        "G001",
        "S274"
      ],
      "before": "exhausted trial divisors below4;terminal remaining=11;d*d>remaining;ORIGINAL n=66",
      "after": "complete increasing prime multiset=[2, 3, 11];full product=66;a composite terminal remainder would have an already tested smaller divisor",
      "relation": "implies",
      "rule_id": "upf_final_prime_remainder_complete_existence_instance",
      "conditions": [],
      "explanation": {
        "zh-Hant": "若remaining>1仍合數，其最小prime因數會在已驗範圍，與除盡不變式矛盾，所以補入的末項為prime。這證本例完整分解；一般存在也可對n作強歸納，合數拆成較小正因數後遞歸。",
        "en": "If a terminal remainder greater than1 were composite,its least prime factor would already have been exhausted,a contradiction.The final entry is therefore prime.This certifies this instance;general existence follows by strong induction,recursively splitting a composite into smaller positive factors."
      },
      "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": "S276",
      "uses": [
        "G001",
        "S275"
      ],
      "before": "Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor66:input=66;COMPLETE prime multiset=[2, 3, 11];distinct primes=[2, 3, 11]",
      "after": "mu=-1;empty product case=否;repeated prime=否",
      "relation": "implies",
      "rule_id": "mobius_value_one_repeated_prime_or_squarefree_parity",
      "conditions": [],
      "explanation": {
        "zh-Hant": "1的空prime多重集給mu1=1；任何prime重複表示square divisor而給0；否則依不同prime個數奇偶給正負。排序不丟重數。",
        "en": "The empty prime multiset for1 gives mu1=1.A repeated prime gives a square divisor and value0;otherwise distinct-prime parity determines the sign.Sorting must not discard multiplicities."
      },
      "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": "S277",
      "uses": [
        "G001",
        "S276"
      ],
      "before": "Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor66 := div(462, 66)",
      "after": "Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor66 = 7",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor66，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor66 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "462",
          "66"
        ],
        "output": "7",
        "label": "Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor66"
      }
    },
    {
      "id": "S278",
      "uses": [
        "G001",
        "S277"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/66):actual positive input=7;check complete integer range1..7",
      "after": "ALL positive divisors INCREASING=[1, 7];not only primes/proper divisors or all integers<=n",
      "relation": "implies",
      "rule_id": "mobinv_ALL_positive_divisors_complete_indexing",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整範圍查整除，保1與n；d|n不是d<=n，也不只取prime或proper divisor。",
        "en": "Check divisibility over the complete range,retaining1 and n.The relation d|n is neither d<=n nor a prime/proper-divisor restriction."
      },
      "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": "S279",
      "uses": [
        "G001",
        "S278"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/66) f(1) ORIGINAL a times divisor := mul(20, 1)",
      "after": "Mobius inversion ORIGINAL inverse F(n/66) f(1) ORIGINAL a times divisor = 20",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/66) f(1) ORIGINAL a times divisor，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/66) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "20",
          "1"
        ],
        "output": "20",
        "label": "Mobius inversion ORIGINAL inverse F(n/66) f(1) ORIGINAL a times divisor"
      }
    },
    {
      "id": "S280",
      "uses": [
        "G001",
        "S279"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/66) f(1) plus ORIGINAL b := sum(20, 1)",
      "after": "Mobius inversion ORIGINAL inverse F(n/66) f(1) plus ORIGINAL b = 21",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/66) f(1) plus ORIGINAL b，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/66) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "20",
          "1"
        ],
        "output": "21",
        "label": "Mobius inversion ORIGINAL inverse F(n/66) f(1) plus ORIGINAL b"
      }
    },
    {
      "id": "S281",
      "uses": [
        "G001",
        "S280"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/66) f(7) ORIGINAL a times divisor := mul(20, 7)",
      "after": "Mobius inversion ORIGINAL inverse F(n/66) f(7) ORIGINAL a times divisor = 140",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/66) f(7) ORIGINAL a times divisor，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/66) f(7) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "20",
          "7"
        ],
        "output": "140",
        "label": "Mobius inversion ORIGINAL inverse F(n/66) f(7) ORIGINAL a times divisor"
      }
    },
    {
      "id": "S282",
      "uses": [
        "G001",
        "S281"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/66) f(7) plus ORIGINAL b := sum(140, 1)",
      "after": "Mobius inversion ORIGINAL inverse F(n/66) f(7) plus ORIGINAL b = 141",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/66) f(7) plus ORIGINAL b，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/66) f(7) plus ORIGINAL b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "140",
          "1"
        ],
        "output": "141",
        "label": "Mobius inversion ORIGINAL inverse F(n/66) f(7) plus ORIGINAL b"
      }
    },
    {
      "id": "S283",
      "uses": [
        "G001",
        "S282"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/66) full forward divisor sum := sum(21, 141)",
      "after": "Mobius inversion ORIGINAL inverse F(n/66) full forward divisor sum = 162",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/66) full forward divisor sum，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/66) full forward divisor sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "21",
          "141"
        ],
        "output": "162",
        "label": "Mobius inversion ORIGINAL inverse F(n/66) full forward divisor sum"
      }
    },
    {
      "id": "S284",
      "uses": [
        "G001",
        "S283"
      ],
      "before": "Mobius inversion ORIGINAL inverse mu(66) times SAME F(n/66) := mul(-1, 162)",
      "after": "Mobius inversion ORIGINAL inverse mu(66) times SAME F(n/66) = -162",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse mu(66) times SAME F(n/66)，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse mu(66) times SAME F(n/66) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-1",
          "162"
        ],
        "output": "-162",
        "label": "Mobius inversion ORIGINAL inverse mu(66) times SAME F(n/66)"
      }
    },
    {
      "id": "S285",
      "uses": [
        "G001",
        "S284"
      ],
      "before": "Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor77:actual positive input for mu=77",
      "after": "complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search",
      "relation": "implies",
      "rule_id": "mobius_complete_UPF_input_role",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先明示正在算原n還是其正因數的mu；每次需要完整prime重數，不能用找到的幾個因數猜squarefree。",
        "en": "Identify whether the input is the original n or one of its positive divisors.Every mu evaluation requires complete prime multiplicities,not a partial factor search."
      },
      "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": "S286",
      "uses": [
        "G001",
        "S285"
      ],
      "before": "ORIGINAL positive integer absolute magnitude=77>1",
      "after": "initialize remaining=n,prime multiset empty;invariant product(extracted)*remaining=n;every smaller trial divisor has been exhausted",
      "relation": "implies",
      "rule_id": "upf_factorization_remaining_product_invariant",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每次只除真正整除的d，並以已窮盡更小因數維持d為prime的理由，不先假定原n為prime。",
        "en": "Divide only by an actual divisor and exhaust smaller candidates,which justifies primality of each extracted divisor without assuming n is prime."
      },
      "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": "S287",
      "uses": [
        "G001",
        "S286"
      ],
      "before": "next divisor=2;previous remaining=77;all smaller divisors exhausted",
      "after": "extracted multiplicity=0;next remaining=77;remaining modulo2=1;extracted multiset=[]",
      "relation": "implies",
      "rule_id": "upf_complete_trial_division_remaining_invariant",
      "conditions": [],
      "explanation": {
        "zh-Hant": "除盡同一prime的所有重數才前進；不把n的原平方根與remaining的停止責任混淆，也不漏最後的prime remainder。",
        "en": "Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted."
      },
      "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": "S288",
      "uses": [
        "G001",
        "S287"
      ],
      "before": "next divisor=3;previous remaining=77;all smaller divisors exhausted",
      "after": "extracted multiplicity=0;next remaining=77;remaining modulo3=2;extracted multiset=[]",
      "relation": "implies",
      "rule_id": "upf_complete_trial_division_remaining_invariant",
      "conditions": [],
      "explanation": {
        "zh-Hant": "除盡同一prime的所有重數才前進；不把n的原平方根與remaining的停止責任混淆，也不漏最後的prime remainder。",
        "en": "Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted."
      },
      "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": "S289",
      "uses": [
        "G001",
        "S288"
      ],
      "before": "next divisor=4;previous remaining=77;all smaller divisors exhausted",
      "after": "extracted multiplicity=0;next remaining=77;remaining modulo4=1;extracted multiset=[]",
      "relation": "implies",
      "rule_id": "upf_complete_trial_division_remaining_invariant",
      "conditions": [],
      "explanation": {
        "zh-Hant": "除盡同一prime的所有重數才前進；不把n的原平方根與remaining的停止責任混淆，也不漏最後的prime remainder。",
        "en": "Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted."
      },
      "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": "S290",
      "uses": [
        "G001",
        "S289"
      ],
      "before": "next divisor=5;previous remaining=77;all smaller divisors exhausted",
      "after": "extracted multiplicity=0;next remaining=77;remaining modulo5=2;extracted multiset=[]",
      "relation": "implies",
      "rule_id": "upf_complete_trial_division_remaining_invariant",
      "conditions": [],
      "explanation": {
        "zh-Hant": "除盡同一prime的所有重數才前進；不把n的原平方根與remaining的停止責任混淆，也不漏最後的prime remainder。",
        "en": "Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted."
      },
      "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": "S291",
      "uses": [
        "G001",
        "S290"
      ],
      "before": "next divisor=6;previous remaining=77;all smaller divisors exhausted",
      "after": "extracted multiplicity=0;next remaining=77;remaining modulo6=5;extracted multiset=[]",
      "relation": "implies",
      "rule_id": "upf_complete_trial_division_remaining_invariant",
      "conditions": [],
      "explanation": {
        "zh-Hant": "除盡同一prime的所有重數才前進；不把n的原平方根與remaining的停止責任混淆，也不漏最後的prime remainder。",
        "en": "Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted."
      },
      "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": "S292",
      "uses": [
        "G001",
        "S291"
      ],
      "before": "upf extract divisor7 occurrence0 := div(77, 7)",
      "after": "upf extract divisor7 occurrence0 = 11",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。",
        "en": "Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 upf extract divisor7 occurrence0，不以小數近似替代。",
          "en": "Recompute upf extract divisor7 occurrence0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "77",
          "7"
        ],
        "output": "11",
        "label": "upf extract divisor7 occurrence0"
      }
    },
    {
      "id": "S293",
      "uses": [
        "G001",
        "S292"
      ],
      "before": "next divisor=7;previous remaining=77;all smaller divisors exhausted",
      "after": "extracted multiplicity=1;next remaining=11;remaining modulo7=4;extracted multiset=[7]",
      "relation": "implies",
      "rule_id": "upf_complete_trial_division_remaining_invariant",
      "conditions": [],
      "explanation": {
        "zh-Hant": "除盡同一prime的所有重數才前進；不把n的原平方根與remaining的停止責任混淆，也不漏最後的prime remainder。",
        "en": "Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted."
      },
      "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": "S294",
      "uses": [
        "G001",
        "S293"
      ],
      "before": "upf canonical complete product factor0 := mul(1, 7)",
      "after": "upf canonical complete product factor0 = 7",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。",
        "en": "Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 upf canonical complete product factor0，不以小數近似替代。",
          "en": "Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "7"
        ],
        "output": "7",
        "label": "upf canonical complete product factor0"
      }
    },
    {
      "id": "S295",
      "uses": [
        "G001",
        "S294"
      ],
      "before": "upf canonical complete product factor1 := mul(7, 11)",
      "after": "upf canonical complete product factor1 = 77",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。",
        "en": "Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 upf canonical complete product factor1，不以小數近似替代。",
          "en": "Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "7",
          "11"
        ],
        "output": "77",
        "label": "upf canonical complete product factor1"
      }
    },
    {
      "id": "S296",
      "uses": [
        "G001",
        "S295"
      ],
      "before": "exhausted trial divisors below8;terminal remaining=11;d*d>remaining;ORIGINAL n=77",
      "after": "complete increasing prime multiset=[7, 11];full product=77;a composite terminal remainder would have an already tested smaller divisor",
      "relation": "implies",
      "rule_id": "upf_final_prime_remainder_complete_existence_instance",
      "conditions": [],
      "explanation": {
        "zh-Hant": "若remaining>1仍合數，其最小prime因數會在已驗範圍，與除盡不變式矛盾，所以補入的末項為prime。這證本例完整分解；一般存在也可對n作強歸納，合數拆成較小正因數後遞歸。",
        "en": "If a terminal remainder greater than1 were composite,its least prime factor would already have been exhausted,a contradiction.The final entry is therefore prime.This certifies this instance;general existence follows by strong induction,recursively splitting a composite into smaller positive factors."
      },
      "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": "S297",
      "uses": [
        "G001",
        "S296"
      ],
      "before": "Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor77:input=77;COMPLETE prime multiset=[7, 11];distinct primes=[7, 11]",
      "after": "mu=1;empty product case=否;repeated prime=否",
      "relation": "implies",
      "rule_id": "mobius_value_one_repeated_prime_or_squarefree_parity",
      "conditions": [],
      "explanation": {
        "zh-Hant": "1的空prime多重集給mu1=1；任何prime重複表示square divisor而給0；否則依不同prime個數奇偶給正負。排序不丟重數。",
        "en": "The empty prime multiset for1 gives mu1=1.A repeated prime gives a square divisor and value0;otherwise distinct-prime parity determines the sign.Sorting must not discard multiplicities."
      },
      "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": "S298",
      "uses": [
        "G001",
        "S297"
      ],
      "before": "Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor77 := div(462, 77)",
      "after": "Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor77 = 6",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor77，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor77 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "462",
          "77"
        ],
        "output": "6",
        "label": "Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor77"
      }
    },
    {
      "id": "S299",
      "uses": [
        "G001",
        "S298"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/77):actual positive input=6;check complete integer range1..6",
      "after": "ALL positive divisors INCREASING=[1, 2, 3, 6];not only primes/proper divisors or all integers<=n",
      "relation": "implies",
      "rule_id": "mobinv_ALL_positive_divisors_complete_indexing",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整範圍查整除，保1與n；d|n不是d<=n，也不只取prime或proper divisor。",
        "en": "Check divisibility over the complete range,retaining1 and n.The relation d|n is neither d<=n nor a prime/proper-divisor restriction."
      },
      "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": "S300",
      "uses": [
        "G001",
        "S299"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/77) f(1) ORIGINAL a times divisor := mul(20, 1)",
      "after": "Mobius inversion ORIGINAL inverse F(n/77) f(1) ORIGINAL a times divisor = 20",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/77) f(1) ORIGINAL a times divisor，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/77) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "20",
          "1"
        ],
        "output": "20",
        "label": "Mobius inversion ORIGINAL inverse F(n/77) f(1) ORIGINAL a times divisor"
      }
    },
    {
      "id": "S301",
      "uses": [
        "G001",
        "S300"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/77) f(1) plus ORIGINAL b := sum(20, 1)",
      "after": "Mobius inversion ORIGINAL inverse F(n/77) f(1) plus ORIGINAL b = 21",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/77) f(1) plus ORIGINAL b，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/77) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "20",
          "1"
        ],
        "output": "21",
        "label": "Mobius inversion ORIGINAL inverse F(n/77) f(1) plus ORIGINAL b"
      }
    },
    {
      "id": "S302",
      "uses": [
        "G001",
        "S301"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/77) f(2) ORIGINAL a times divisor := mul(20, 2)",
      "after": "Mobius inversion ORIGINAL inverse F(n/77) f(2) ORIGINAL a times divisor = 40",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/77) f(2) ORIGINAL a times divisor，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/77) f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "20",
          "2"
        ],
        "output": "40",
        "label": "Mobius inversion ORIGINAL inverse F(n/77) f(2) ORIGINAL a times divisor"
      }
    },
    {
      "id": "S303",
      "uses": [
        "G001",
        "S302"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/77) f(2) plus ORIGINAL b := sum(40, 1)",
      "after": "Mobius inversion ORIGINAL inverse F(n/77) f(2) plus ORIGINAL b = 41",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/77) f(2) plus ORIGINAL b，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/77) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "40",
          "1"
        ],
        "output": "41",
        "label": "Mobius inversion ORIGINAL inverse F(n/77) f(2) plus ORIGINAL b"
      }
    },
    {
      "id": "S304",
      "uses": [
        "G001",
        "S303"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/77) f(3) ORIGINAL a times divisor := mul(20, 3)",
      "after": "Mobius inversion ORIGINAL inverse F(n/77) f(3) ORIGINAL a times divisor = 60",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/77) f(3) ORIGINAL a times divisor，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/77) f(3) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "20",
          "3"
        ],
        "output": "60",
        "label": "Mobius inversion ORIGINAL inverse F(n/77) f(3) ORIGINAL a times divisor"
      }
    },
    {
      "id": "S305",
      "uses": [
        "G001",
        "S304"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/77) f(3) plus ORIGINAL b := sum(60, 1)",
      "after": "Mobius inversion ORIGINAL inverse F(n/77) f(3) plus ORIGINAL b = 61",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/77) f(3) plus ORIGINAL b，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/77) f(3) plus ORIGINAL b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "60",
          "1"
        ],
        "output": "61",
        "label": "Mobius inversion ORIGINAL inverse F(n/77) f(3) plus ORIGINAL b"
      }
    },
    {
      "id": "S306",
      "uses": [
        "G001",
        "S305"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/77) f(6) ORIGINAL a times divisor := mul(20, 6)",
      "after": "Mobius inversion ORIGINAL inverse F(n/77) f(6) ORIGINAL a times divisor = 120",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/77) f(6) ORIGINAL a times divisor，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/77) f(6) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "20",
          "6"
        ],
        "output": "120",
        "label": "Mobius inversion ORIGINAL inverse F(n/77) f(6) ORIGINAL a times divisor"
      }
    },
    {
      "id": "S307",
      "uses": [
        "G001",
        "S306"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/77) f(6) plus ORIGINAL b := sum(120, 1)",
      "after": "Mobius inversion ORIGINAL inverse F(n/77) f(6) plus ORIGINAL b = 121",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/77) f(6) plus ORIGINAL b，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/77) f(6) plus ORIGINAL b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "120",
          "1"
        ],
        "output": "121",
        "label": "Mobius inversion ORIGINAL inverse F(n/77) f(6) plus ORIGINAL b"
      }
    },
    {
      "id": "S308",
      "uses": [
        "G001",
        "S307"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/77) full forward divisor sum := sum(21, 41, 61, 121)",
      "after": "Mobius inversion ORIGINAL inverse F(n/77) full forward divisor sum = 244",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/77) full forward divisor sum，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/77) full forward divisor sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "21",
          "41",
          "61",
          "121"
        ],
        "output": "244",
        "label": "Mobius inversion ORIGINAL inverse F(n/77) full forward divisor sum"
      }
    },
    {
      "id": "S309",
      "uses": [
        "G001",
        "S308"
      ],
      "before": "Mobius inversion ORIGINAL inverse mu(77) times SAME F(n/77) := mul(1, 244)",
      "after": "Mobius inversion ORIGINAL inverse mu(77) times SAME F(n/77) = 244",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse mu(77) times SAME F(n/77)，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse mu(77) times SAME F(n/77) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "244"
        ],
        "output": "244",
        "label": "Mobius inversion ORIGINAL inverse mu(77) times SAME F(n/77)"
      }
    },
    {
      "id": "S310",
      "uses": [
        "G001",
        "S309"
      ],
      "before": "Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor154:actual positive input for mu=154",
      "after": "complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search",
      "relation": "implies",
      "rule_id": "mobius_complete_UPF_input_role",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先明示正在算原n還是其正因數的mu；每次需要完整prime重數，不能用找到的幾個因數猜squarefree。",
        "en": "Identify whether the input is the original n or one of its positive divisors.Every mu evaluation requires complete prime multiplicities,not a partial factor search."
      },
      "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": "S311",
      "uses": [
        "G001",
        "S310"
      ],
      "before": "ORIGINAL positive integer absolute magnitude=154>1",
      "after": "initialize remaining=n,prime multiset empty;invariant product(extracted)*remaining=n;every smaller trial divisor has been exhausted",
      "relation": "implies",
      "rule_id": "upf_factorization_remaining_product_invariant",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每次只除真正整除的d，並以已窮盡更小因數維持d為prime的理由，不先假定原n為prime。",
        "en": "Divide only by an actual divisor and exhaust smaller candidates,which justifies primality of each extracted divisor without assuming n is prime."
      },
      "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": "S312",
      "uses": [
        "G001",
        "S311"
      ],
      "before": "upf extract divisor2 occurrence0 := div(154, 2)",
      "after": "upf extract divisor2 occurrence0 = 77",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。",
        "en": "Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 upf extract divisor2 occurrence0，不以小數近似替代。",
          "en": "Recompute upf extract divisor2 occurrence0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "154",
          "2"
        ],
        "output": "77",
        "label": "upf extract divisor2 occurrence0"
      }
    },
    {
      "id": "S313",
      "uses": [
        "G001",
        "S312"
      ],
      "before": "next divisor=2;previous remaining=154;all smaller divisors exhausted",
      "after": "extracted multiplicity=1;next remaining=77;remaining modulo2=1;extracted multiset=[2]",
      "relation": "implies",
      "rule_id": "upf_complete_trial_division_remaining_invariant",
      "conditions": [],
      "explanation": {
        "zh-Hant": "除盡同一prime的所有重數才前進；不把n的原平方根與remaining的停止責任混淆，也不漏最後的prime remainder。",
        "en": "Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted."
      },
      "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": "S314",
      "uses": [
        "G001",
        "S313"
      ],
      "before": "next divisor=3;previous remaining=77;all smaller divisors exhausted",
      "after": "extracted multiplicity=0;next remaining=77;remaining modulo3=2;extracted multiset=[2]",
      "relation": "implies",
      "rule_id": "upf_complete_trial_division_remaining_invariant",
      "conditions": [],
      "explanation": {
        "zh-Hant": "除盡同一prime的所有重數才前進；不把n的原平方根與remaining的停止責任混淆，也不漏最後的prime remainder。",
        "en": "Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted."
      },
      "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": "S315",
      "uses": [
        "G001",
        "S314"
      ],
      "before": "next divisor=4;previous remaining=77;all smaller divisors exhausted",
      "after": "extracted multiplicity=0;next remaining=77;remaining modulo4=1;extracted multiset=[2]",
      "relation": "implies",
      "rule_id": "upf_complete_trial_division_remaining_invariant",
      "conditions": [],
      "explanation": {
        "zh-Hant": "除盡同一prime的所有重數才前進；不把n的原平方根與remaining的停止責任混淆，也不漏最後的prime remainder。",
        "en": "Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted."
      },
      "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": "S316",
      "uses": [
        "G001",
        "S315"
      ],
      "before": "next divisor=5;previous remaining=77;all smaller divisors exhausted",
      "after": "extracted multiplicity=0;next remaining=77;remaining modulo5=2;extracted multiset=[2]",
      "relation": "implies",
      "rule_id": "upf_complete_trial_division_remaining_invariant",
      "conditions": [],
      "explanation": {
        "zh-Hant": "除盡同一prime的所有重數才前進；不把n的原平方根與remaining的停止責任混淆，也不漏最後的prime remainder。",
        "en": "Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted."
      },
      "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": "S317",
      "uses": [
        "G001",
        "S316"
      ],
      "before": "next divisor=6;previous remaining=77;all smaller divisors exhausted",
      "after": "extracted multiplicity=0;next remaining=77;remaining modulo6=5;extracted multiset=[2]",
      "relation": "implies",
      "rule_id": "upf_complete_trial_division_remaining_invariant",
      "conditions": [],
      "explanation": {
        "zh-Hant": "除盡同一prime的所有重數才前進；不把n的原平方根與remaining的停止責任混淆，也不漏最後的prime remainder。",
        "en": "Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted."
      },
      "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": "S318",
      "uses": [
        "G001",
        "S317"
      ],
      "before": "upf extract divisor7 occurrence1 := div(77, 7)",
      "after": "upf extract divisor7 occurrence1 = 11",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。",
        "en": "Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 upf extract divisor7 occurrence1，不以小數近似替代。",
          "en": "Recompute upf extract divisor7 occurrence1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "77",
          "7"
        ],
        "output": "11",
        "label": "upf extract divisor7 occurrence1"
      }
    },
    {
      "id": "S319",
      "uses": [
        "G001",
        "S318"
      ],
      "before": "next divisor=7;previous remaining=77;all smaller divisors exhausted",
      "after": "extracted multiplicity=1;next remaining=11;remaining modulo7=4;extracted multiset=[2, 7]",
      "relation": "implies",
      "rule_id": "upf_complete_trial_division_remaining_invariant",
      "conditions": [],
      "explanation": {
        "zh-Hant": "除盡同一prime的所有重數才前進；不把n的原平方根與remaining的停止責任混淆，也不漏最後的prime remainder。",
        "en": "Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted."
      },
      "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": "S320",
      "uses": [
        "G001",
        "S319"
      ],
      "before": "upf canonical complete product factor0 := mul(1, 2)",
      "after": "upf canonical complete product factor0 = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。",
        "en": "Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 upf canonical complete product factor0，不以小數近似替代。",
          "en": "Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "2"
        ],
        "output": "2",
        "label": "upf canonical complete product factor0"
      }
    },
    {
      "id": "S321",
      "uses": [
        "G001",
        "S320"
      ],
      "before": "upf canonical complete product factor1 := mul(2, 7)",
      "after": "upf canonical complete product factor1 = 14",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。",
        "en": "Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 upf canonical complete product factor1，不以小數近似替代。",
          "en": "Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "7"
        ],
        "output": "14",
        "label": "upf canonical complete product factor1"
      }
    },
    {
      "id": "S322",
      "uses": [
        "G001",
        "S321"
      ],
      "before": "upf canonical complete product factor2 := mul(14, 11)",
      "after": "upf canonical complete product factor2 = 154",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。",
        "en": "Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 upf canonical complete product factor2，不以小數近似替代。",
          "en": "Recompute upf canonical complete product factor2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "14",
          "11"
        ],
        "output": "154",
        "label": "upf canonical complete product factor2"
      }
    },
    {
      "id": "S323",
      "uses": [
        "G001",
        "S322"
      ],
      "before": "exhausted trial divisors below8;terminal remaining=11;d*d>remaining;ORIGINAL n=154",
      "after": "complete increasing prime multiset=[2, 7, 11];full product=154;a composite terminal remainder would have an already tested smaller divisor",
      "relation": "implies",
      "rule_id": "upf_final_prime_remainder_complete_existence_instance",
      "conditions": [],
      "explanation": {
        "zh-Hant": "若remaining>1仍合數，其最小prime因數會在已驗範圍，與除盡不變式矛盾，所以補入的末項為prime。這證本例完整分解；一般存在也可對n作強歸納，合數拆成較小正因數後遞歸。",
        "en": "If a terminal remainder greater than1 were composite,its least prime factor would already have been exhausted,a contradiction.The final entry is therefore prime.This certifies this instance;general existence follows by strong induction,recursively splitting a composite into smaller positive factors."
      },
      "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": "S324",
      "uses": [
        "G001",
        "S323"
      ],
      "before": "Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor154:input=154;COMPLETE prime multiset=[2, 7, 11];distinct primes=[2, 7, 11]",
      "after": "mu=-1;empty product case=否;repeated prime=否",
      "relation": "implies",
      "rule_id": "mobius_value_one_repeated_prime_or_squarefree_parity",
      "conditions": [],
      "explanation": {
        "zh-Hant": "1的空prime多重集給mu1=1；任何prime重複表示square divisor而給0；否則依不同prime個數奇偶給正負。排序不丟重數。",
        "en": "The empty prime multiset for1 gives mu1=1.A repeated prime gives a square divisor and value0;otherwise distinct-prime parity determines the sign.Sorting must not discard multiplicities."
      },
      "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": "S325",
      "uses": [
        "G001",
        "S324"
      ],
      "before": "Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor154 := div(462, 154)",
      "after": "Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor154 = 3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor154，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor154 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "462",
          "154"
        ],
        "output": "3",
        "label": "Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor154"
      }
    },
    {
      "id": "S326",
      "uses": [
        "G001",
        "S325"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/154):actual positive input=3;check complete integer range1..3",
      "after": "ALL positive divisors INCREASING=[1, 3];not only primes/proper divisors or all integers<=n",
      "relation": "implies",
      "rule_id": "mobinv_ALL_positive_divisors_complete_indexing",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整範圍查整除，保1與n；d|n不是d<=n，也不只取prime或proper divisor。",
        "en": "Check divisibility over the complete range,retaining1 and n.The relation d|n is neither d<=n nor a prime/proper-divisor restriction."
      },
      "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": "S327",
      "uses": [
        "G001",
        "S326"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/154) f(1) ORIGINAL a times divisor := mul(20, 1)",
      "after": "Mobius inversion ORIGINAL inverse F(n/154) f(1) ORIGINAL a times divisor = 20",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/154) f(1) ORIGINAL a times divisor，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/154) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "20",
          "1"
        ],
        "output": "20",
        "label": "Mobius inversion ORIGINAL inverse F(n/154) f(1) ORIGINAL a times divisor"
      }
    },
    {
      "id": "S328",
      "uses": [
        "G001",
        "S327"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/154) f(1) plus ORIGINAL b := sum(20, 1)",
      "after": "Mobius inversion ORIGINAL inverse F(n/154) f(1) plus ORIGINAL b = 21",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/154) f(1) plus ORIGINAL b，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/154) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "20",
          "1"
        ],
        "output": "21",
        "label": "Mobius inversion ORIGINAL inverse F(n/154) f(1) plus ORIGINAL b"
      }
    },
    {
      "id": "S329",
      "uses": [
        "G001",
        "S328"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/154) f(3) ORIGINAL a times divisor := mul(20, 3)",
      "after": "Mobius inversion ORIGINAL inverse F(n/154) f(3) ORIGINAL a times divisor = 60",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/154) f(3) ORIGINAL a times divisor，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/154) f(3) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "20",
          "3"
        ],
        "output": "60",
        "label": "Mobius inversion ORIGINAL inverse F(n/154) f(3) ORIGINAL a times divisor"
      }
    },
    {
      "id": "S330",
      "uses": [
        "G001",
        "S329"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/154) f(3) plus ORIGINAL b := sum(60, 1)",
      "after": "Mobius inversion ORIGINAL inverse F(n/154) f(3) plus ORIGINAL b = 61",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/154) f(3) plus ORIGINAL b，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/154) f(3) plus ORIGINAL b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "60",
          "1"
        ],
        "output": "61",
        "label": "Mobius inversion ORIGINAL inverse F(n/154) f(3) plus ORIGINAL b"
      }
    },
    {
      "id": "S331",
      "uses": [
        "G001",
        "S330"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/154) full forward divisor sum := sum(21, 61)",
      "after": "Mobius inversion ORIGINAL inverse F(n/154) full forward divisor sum = 82",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/154) full forward divisor sum，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/154) full forward divisor sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "21",
          "61"
        ],
        "output": "82",
        "label": "Mobius inversion ORIGINAL inverse F(n/154) full forward divisor sum"
      }
    },
    {
      "id": "S332",
      "uses": [
        "G001",
        "S331"
      ],
      "before": "Mobius inversion ORIGINAL inverse mu(154) times SAME F(n/154) := mul(-1, 82)",
      "after": "Mobius inversion ORIGINAL inverse mu(154) times SAME F(n/154) = -82",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse mu(154) times SAME F(n/154)，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse mu(154) times SAME F(n/154) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-1",
          "82"
        ],
        "output": "-82",
        "label": "Mobius inversion ORIGINAL inverse mu(154) times SAME F(n/154)"
      }
    },
    {
      "id": "S333",
      "uses": [
        "G001",
        "S332"
      ],
      "before": "Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor231:actual positive input for mu=231",
      "after": "complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search",
      "relation": "implies",
      "rule_id": "mobius_complete_UPF_input_role",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先明示正在算原n還是其正因數的mu；每次需要完整prime重數，不能用找到的幾個因數猜squarefree。",
        "en": "Identify whether the input is the original n or one of its positive divisors.Every mu evaluation requires complete prime multiplicities,not a partial factor search."
      },
      "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": "S334",
      "uses": [
        "G001",
        "S333"
      ],
      "before": "ORIGINAL positive integer absolute magnitude=231>1",
      "after": "initialize remaining=n,prime multiset empty;invariant product(extracted)*remaining=n;every smaller trial divisor has been exhausted",
      "relation": "implies",
      "rule_id": "upf_factorization_remaining_product_invariant",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每次只除真正整除的d，並以已窮盡更小因數維持d為prime的理由，不先假定原n為prime。",
        "en": "Divide only by an actual divisor and exhaust smaller candidates,which justifies primality of each extracted divisor without assuming n is prime."
      },
      "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": "S335",
      "uses": [
        "G001",
        "S334"
      ],
      "before": "next divisor=2;previous remaining=231;all smaller divisors exhausted",
      "after": "extracted multiplicity=0;next remaining=231;remaining modulo2=1;extracted multiset=[]",
      "relation": "implies",
      "rule_id": "upf_complete_trial_division_remaining_invariant",
      "conditions": [],
      "explanation": {
        "zh-Hant": "除盡同一prime的所有重數才前進；不把n的原平方根與remaining的停止責任混淆，也不漏最後的prime remainder。",
        "en": "Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted."
      },
      "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": "S336",
      "uses": [
        "G001",
        "S335"
      ],
      "before": "upf extract divisor3 occurrence0 := div(231, 3)",
      "after": "upf extract divisor3 occurrence0 = 77",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。",
        "en": "Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 upf extract divisor3 occurrence0，不以小數近似替代。",
          "en": "Recompute upf extract divisor3 occurrence0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "231",
          "3"
        ],
        "output": "77",
        "label": "upf extract divisor3 occurrence0"
      }
    },
    {
      "id": "S337",
      "uses": [
        "G001",
        "S336"
      ],
      "before": "next divisor=3;previous remaining=231;all smaller divisors exhausted",
      "after": "extracted multiplicity=1;next remaining=77;remaining modulo3=2;extracted multiset=[3]",
      "relation": "implies",
      "rule_id": "upf_complete_trial_division_remaining_invariant",
      "conditions": [],
      "explanation": {
        "zh-Hant": "除盡同一prime的所有重數才前進；不把n的原平方根與remaining的停止責任混淆，也不漏最後的prime remainder。",
        "en": "Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted."
      },
      "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": "S338",
      "uses": [
        "G001",
        "S337"
      ],
      "before": "next divisor=4;previous remaining=77;all smaller divisors exhausted",
      "after": "extracted multiplicity=0;next remaining=77;remaining modulo4=1;extracted multiset=[3]",
      "relation": "implies",
      "rule_id": "upf_complete_trial_division_remaining_invariant",
      "conditions": [],
      "explanation": {
        "zh-Hant": "除盡同一prime的所有重數才前進；不把n的原平方根與remaining的停止責任混淆，也不漏最後的prime remainder。",
        "en": "Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted."
      },
      "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": "S339",
      "uses": [
        "G001",
        "S338"
      ],
      "before": "next divisor=5;previous remaining=77;all smaller divisors exhausted",
      "after": "extracted multiplicity=0;next remaining=77;remaining modulo5=2;extracted multiset=[3]",
      "relation": "implies",
      "rule_id": "upf_complete_trial_division_remaining_invariant",
      "conditions": [],
      "explanation": {
        "zh-Hant": "除盡同一prime的所有重數才前進；不把n的原平方根與remaining的停止責任混淆，也不漏最後的prime remainder。",
        "en": "Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted."
      },
      "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": "S340",
      "uses": [
        "G001",
        "S339"
      ],
      "before": "next divisor=6;previous remaining=77;all smaller divisors exhausted",
      "after": "extracted multiplicity=0;next remaining=77;remaining modulo6=5;extracted multiset=[3]",
      "relation": "implies",
      "rule_id": "upf_complete_trial_division_remaining_invariant",
      "conditions": [],
      "explanation": {
        "zh-Hant": "除盡同一prime的所有重數才前進；不把n的原平方根與remaining的停止責任混淆，也不漏最後的prime remainder。",
        "en": "Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted."
      },
      "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": "S341",
      "uses": [
        "G001",
        "S340"
      ],
      "before": "upf extract divisor7 occurrence1 := div(77, 7)",
      "after": "upf extract divisor7 occurrence1 = 11",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。",
        "en": "Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 upf extract divisor7 occurrence1，不以小數近似替代。",
          "en": "Recompute upf extract divisor7 occurrence1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "77",
          "7"
        ],
        "output": "11",
        "label": "upf extract divisor7 occurrence1"
      }
    },
    {
      "id": "S342",
      "uses": [
        "G001",
        "S341"
      ],
      "before": "next divisor=7;previous remaining=77;all smaller divisors exhausted",
      "after": "extracted multiplicity=1;next remaining=11;remaining modulo7=4;extracted multiset=[3, 7]",
      "relation": "implies",
      "rule_id": "upf_complete_trial_division_remaining_invariant",
      "conditions": [],
      "explanation": {
        "zh-Hant": "除盡同一prime的所有重數才前進；不把n的原平方根與remaining的停止責任混淆，也不漏最後的prime remainder。",
        "en": "Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted."
      },
      "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": "S343",
      "uses": [
        "G001",
        "S342"
      ],
      "before": "upf canonical complete product factor0 := mul(1, 3)",
      "after": "upf canonical complete product factor0 = 3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。",
        "en": "Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 upf canonical complete product factor0，不以小數近似替代。",
          "en": "Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "3"
        ],
        "output": "3",
        "label": "upf canonical complete product factor0"
      }
    },
    {
      "id": "S344",
      "uses": [
        "G001",
        "S343"
      ],
      "before": "upf canonical complete product factor1 := mul(3, 7)",
      "after": "upf canonical complete product factor1 = 21",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。",
        "en": "Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 upf canonical complete product factor1，不以小數近似替代。",
          "en": "Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "3",
          "7"
        ],
        "output": "21",
        "label": "upf canonical complete product factor1"
      }
    },
    {
      "id": "S345",
      "uses": [
        "G001",
        "S344"
      ],
      "before": "upf canonical complete product factor2 := mul(21, 11)",
      "after": "upf canonical complete product factor2 = 231",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。",
        "en": "Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 upf canonical complete product factor2，不以小數近似替代。",
          "en": "Recompute upf canonical complete product factor2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "21",
          "11"
        ],
        "output": "231",
        "label": "upf canonical complete product factor2"
      }
    },
    {
      "id": "S346",
      "uses": [
        "G001",
        "S345"
      ],
      "before": "exhausted trial divisors below8;terminal remaining=11;d*d>remaining;ORIGINAL n=231",
      "after": "complete increasing prime multiset=[3, 7, 11];full product=231;a composite terminal remainder would have an already tested smaller divisor",
      "relation": "implies",
      "rule_id": "upf_final_prime_remainder_complete_existence_instance",
      "conditions": [],
      "explanation": {
        "zh-Hant": "若remaining>1仍合數，其最小prime因數會在已驗範圍，與除盡不變式矛盾，所以補入的末項為prime。這證本例完整分解；一般存在也可對n作強歸納，合數拆成較小正因數後遞歸。",
        "en": "If a terminal remainder greater than1 were composite,its least prime factor would already have been exhausted,a contradiction.The final entry is therefore prime.This certifies this instance;general existence follows by strong induction,recursively splitting a composite into smaller positive factors."
      },
      "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": "S347",
      "uses": [
        "G001",
        "S346"
      ],
      "before": "Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor231:input=231;COMPLETE prime multiset=[3, 7, 11];distinct primes=[3, 7, 11]",
      "after": "mu=-1;empty product case=否;repeated prime=否",
      "relation": "implies",
      "rule_id": "mobius_value_one_repeated_prime_or_squarefree_parity",
      "conditions": [],
      "explanation": {
        "zh-Hant": "1的空prime多重集給mu1=1；任何prime重複表示square divisor而給0；否則依不同prime個數奇偶給正負。排序不丟重數。",
        "en": "The empty prime multiset for1 gives mu1=1.A repeated prime gives a square divisor and value0;otherwise distinct-prime parity determines the sign.Sorting must not discard multiplicities."
      },
      "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": "S348",
      "uses": [
        "G001",
        "S347"
      ],
      "before": "Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor231 := div(462, 231)",
      "after": "Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor231 = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor231，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor231 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "462",
          "231"
        ],
        "output": "2",
        "label": "Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor231"
      }
    },
    {
      "id": "S349",
      "uses": [
        "G001",
        "S348"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/231):actual positive input=2;check complete integer range1..2",
      "after": "ALL positive divisors INCREASING=[1, 2];not only primes/proper divisors or all integers<=n",
      "relation": "implies",
      "rule_id": "mobinv_ALL_positive_divisors_complete_indexing",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整範圍查整除，保1與n；d|n不是d<=n，也不只取prime或proper divisor。",
        "en": "Check divisibility over the complete range,retaining1 and n.The relation d|n is neither d<=n nor a prime/proper-divisor restriction."
      },
      "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": "S350",
      "uses": [
        "G001",
        "S349"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/231) f(1) ORIGINAL a times divisor := mul(20, 1)",
      "after": "Mobius inversion ORIGINAL inverse F(n/231) f(1) ORIGINAL a times divisor = 20",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/231) f(1) ORIGINAL a times divisor，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/231) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "20",
          "1"
        ],
        "output": "20",
        "label": "Mobius inversion ORIGINAL inverse F(n/231) f(1) ORIGINAL a times divisor"
      }
    },
    {
      "id": "S351",
      "uses": [
        "G001",
        "S350"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/231) f(1) plus ORIGINAL b := sum(20, 1)",
      "after": "Mobius inversion ORIGINAL inverse F(n/231) f(1) plus ORIGINAL b = 21",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/231) f(1) plus ORIGINAL b，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/231) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "20",
          "1"
        ],
        "output": "21",
        "label": "Mobius inversion ORIGINAL inverse F(n/231) f(1) plus ORIGINAL b"
      }
    },
    {
      "id": "S352",
      "uses": [
        "G001",
        "S351"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/231) f(2) ORIGINAL a times divisor := mul(20, 2)",
      "after": "Mobius inversion ORIGINAL inverse F(n/231) f(2) ORIGINAL a times divisor = 40",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/231) f(2) ORIGINAL a times divisor，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/231) f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "20",
          "2"
        ],
        "output": "40",
        "label": "Mobius inversion ORIGINAL inverse F(n/231) f(2) ORIGINAL a times divisor"
      }
    },
    {
      "id": "S353",
      "uses": [
        "G001",
        "S352"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/231) f(2) plus ORIGINAL b := sum(40, 1)",
      "after": "Mobius inversion ORIGINAL inverse F(n/231) f(2) plus ORIGINAL b = 41",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/231) f(2) plus ORIGINAL b，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/231) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "40",
          "1"
        ],
        "output": "41",
        "label": "Mobius inversion ORIGINAL inverse F(n/231) f(2) plus ORIGINAL b"
      }
    },
    {
      "id": "S354",
      "uses": [
        "G001",
        "S353"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/231) full forward divisor sum := sum(21, 41)",
      "after": "Mobius inversion ORIGINAL inverse F(n/231) full forward divisor sum = 62",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/231) full forward divisor sum，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/231) full forward divisor sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "21",
          "41"
        ],
        "output": "62",
        "label": "Mobius inversion ORIGINAL inverse F(n/231) full forward divisor sum"
      }
    },
    {
      "id": "S355",
      "uses": [
        "G001",
        "S354"
      ],
      "before": "Mobius inversion ORIGINAL inverse mu(231) times SAME F(n/231) := mul(-1, 62)",
      "after": "Mobius inversion ORIGINAL inverse mu(231) times SAME F(n/231) = -62",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse mu(231) times SAME F(n/231)，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse mu(231) times SAME F(n/231) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-1",
          "62"
        ],
        "output": "-62",
        "label": "Mobius inversion ORIGINAL inverse mu(231) times SAME F(n/231)"
      }
    },
    {
      "id": "S356",
      "uses": [
        "G001",
        "S355"
      ],
      "before": "Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor462:actual positive input for mu=462",
      "after": "complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search",
      "relation": "implies",
      "rule_id": "mobius_complete_UPF_input_role",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先明示正在算原n還是其正因數的mu；每次需要完整prime重數，不能用找到的幾個因數猜squarefree。",
        "en": "Identify whether the input is the original n or one of its positive divisors.Every mu evaluation requires complete prime multiplicities,not a partial factor search."
      },
      "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": "S357",
      "uses": [
        "G001",
        "S356"
      ],
      "before": "ORIGINAL positive integer absolute magnitude=462>1",
      "after": "initialize remaining=n,prime multiset empty;invariant product(extracted)*remaining=n;every smaller trial divisor has been exhausted",
      "relation": "implies",
      "rule_id": "upf_factorization_remaining_product_invariant",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每次只除真正整除的d，並以已窮盡更小因數維持d為prime的理由，不先假定原n為prime。",
        "en": "Divide only by an actual divisor and exhaust smaller candidates,which justifies primality of each extracted divisor without assuming n is prime."
      },
      "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": "S358",
      "uses": [
        "G001",
        "S357"
      ],
      "before": "upf extract divisor2 occurrence0 := div(462, 2)",
      "after": "upf extract divisor2 occurrence0 = 231",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。",
        "en": "Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 upf extract divisor2 occurrence0，不以小數近似替代。",
          "en": "Recompute upf extract divisor2 occurrence0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "462",
          "2"
        ],
        "output": "231",
        "label": "upf extract divisor2 occurrence0"
      }
    },
    {
      "id": "S359",
      "uses": [
        "G001",
        "S358"
      ],
      "before": "next divisor=2;previous remaining=462;all smaller divisors exhausted",
      "after": "extracted multiplicity=1;next remaining=231;remaining modulo2=1;extracted multiset=[2]",
      "relation": "implies",
      "rule_id": "upf_complete_trial_division_remaining_invariant",
      "conditions": [],
      "explanation": {
        "zh-Hant": "除盡同一prime的所有重數才前進；不把n的原平方根與remaining的停止責任混淆，也不漏最後的prime remainder。",
        "en": "Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted."
      },
      "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": "S360",
      "uses": [
        "G001",
        "S359"
      ],
      "before": "upf extract divisor3 occurrence1 := div(231, 3)",
      "after": "upf extract divisor3 occurrence1 = 77",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。",
        "en": "Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 upf extract divisor3 occurrence1，不以小數近似替代。",
          "en": "Recompute upf extract divisor3 occurrence1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "231",
          "3"
        ],
        "output": "77",
        "label": "upf extract divisor3 occurrence1"
      }
    },
    {
      "id": "S361",
      "uses": [
        "G001",
        "S360"
      ],
      "before": "next divisor=3;previous remaining=231;all smaller divisors exhausted",
      "after": "extracted multiplicity=1;next remaining=77;remaining modulo3=2;extracted multiset=[2, 3]",
      "relation": "implies",
      "rule_id": "upf_complete_trial_division_remaining_invariant",
      "conditions": [],
      "explanation": {
        "zh-Hant": "除盡同一prime的所有重數才前進；不把n的原平方根與remaining的停止責任混淆，也不漏最後的prime remainder。",
        "en": "Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted."
      },
      "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": "S362",
      "uses": [
        "G001",
        "S361"
      ],
      "before": "next divisor=4;previous remaining=77;all smaller divisors exhausted",
      "after": "extracted multiplicity=0;next remaining=77;remaining modulo4=1;extracted multiset=[2, 3]",
      "relation": "implies",
      "rule_id": "upf_complete_trial_division_remaining_invariant",
      "conditions": [],
      "explanation": {
        "zh-Hant": "除盡同一prime的所有重數才前進；不把n的原平方根與remaining的停止責任混淆，也不漏最後的prime remainder。",
        "en": "Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted."
      },
      "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": "S363",
      "uses": [
        "G001",
        "S362"
      ],
      "before": "next divisor=5;previous remaining=77;all smaller divisors exhausted",
      "after": "extracted multiplicity=0;next remaining=77;remaining modulo5=2;extracted multiset=[2, 3]",
      "relation": "implies",
      "rule_id": "upf_complete_trial_division_remaining_invariant",
      "conditions": [],
      "explanation": {
        "zh-Hant": "除盡同一prime的所有重數才前進；不把n的原平方根與remaining的停止責任混淆，也不漏最後的prime remainder。",
        "en": "Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted."
      },
      "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": "S364",
      "uses": [
        "G001",
        "S363"
      ],
      "before": "next divisor=6;previous remaining=77;all smaller divisors exhausted",
      "after": "extracted multiplicity=0;next remaining=77;remaining modulo6=5;extracted multiset=[2, 3]",
      "relation": "implies",
      "rule_id": "upf_complete_trial_division_remaining_invariant",
      "conditions": [],
      "explanation": {
        "zh-Hant": "除盡同一prime的所有重數才前進；不把n的原平方根與remaining的停止責任混淆，也不漏最後的prime remainder。",
        "en": "Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted."
      },
      "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": "S365",
      "uses": [
        "G001",
        "S364"
      ],
      "before": "upf extract divisor7 occurrence2 := div(77, 7)",
      "after": "upf extract divisor7 occurrence2 = 11",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。",
        "en": "Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 upf extract divisor7 occurrence2，不以小數近似替代。",
          "en": "Recompute upf extract divisor7 occurrence2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "77",
          "7"
        ],
        "output": "11",
        "label": "upf extract divisor7 occurrence2"
      }
    },
    {
      "id": "S366",
      "uses": [
        "G001",
        "S365"
      ],
      "before": "next divisor=7;previous remaining=77;all smaller divisors exhausted",
      "after": "extracted multiplicity=1;next remaining=11;remaining modulo7=4;extracted multiset=[2, 3, 7]",
      "relation": "implies",
      "rule_id": "upf_complete_trial_division_remaining_invariant",
      "conditions": [],
      "explanation": {
        "zh-Hant": "除盡同一prime的所有重數才前進；不把n的原平方根與remaining的停止責任混淆，也不漏最後的prime remainder。",
        "en": "Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted."
      },
      "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": "S367",
      "uses": [
        "G001",
        "S366"
      ],
      "before": "upf canonical complete product factor0 := mul(1, 2)",
      "after": "upf canonical complete product factor0 = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。",
        "en": "Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 upf canonical complete product factor0，不以小數近似替代。",
          "en": "Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "2"
        ],
        "output": "2",
        "label": "upf canonical complete product factor0"
      }
    },
    {
      "id": "S368",
      "uses": [
        "G001",
        "S367"
      ],
      "before": "upf canonical complete product factor1 := mul(2, 3)",
      "after": "upf canonical complete product factor1 = 6",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。",
        "en": "Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 upf canonical complete product factor1，不以小數近似替代。",
          "en": "Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "3"
        ],
        "output": "6",
        "label": "upf canonical complete product factor1"
      }
    },
    {
      "id": "S369",
      "uses": [
        "G001",
        "S368"
      ],
      "before": "upf canonical complete product factor2 := mul(6, 7)",
      "after": "upf canonical complete product factor2 = 42",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。",
        "en": "Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 upf canonical complete product factor2，不以小數近似替代。",
          "en": "Recompute upf canonical complete product factor2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "6",
          "7"
        ],
        "output": "42",
        "label": "upf canonical complete product factor2"
      }
    },
    {
      "id": "S370",
      "uses": [
        "G001",
        "S369"
      ],
      "before": "upf canonical complete product factor3 := mul(42, 11)",
      "after": "upf canonical complete product factor3 = 462",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已核中間量精確代值，保留原順序、正負號及量詞條件。",
        "en": "Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 upf canonical complete product factor3，不以小數近似替代。",
          "en": "Recompute upf canonical complete product factor3 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "42",
          "11"
        ],
        "output": "462",
        "label": "upf canonical complete product factor3"
      }
    },
    {
      "id": "S371",
      "uses": [
        "G001",
        "S370"
      ],
      "before": "exhausted trial divisors below8;terminal remaining=11;d*d>remaining;ORIGINAL n=462",
      "after": "complete increasing prime multiset=[2, 3, 7, 11];full product=462;a composite terminal remainder would have an already tested smaller divisor",
      "relation": "implies",
      "rule_id": "upf_final_prime_remainder_complete_existence_instance",
      "conditions": [],
      "explanation": {
        "zh-Hant": "若remaining>1仍合數，其最小prime因數會在已驗範圍，與除盡不變式矛盾，所以補入的末項為prime。這證本例完整分解；一般存在也可對n作強歸納，合數拆成較小正因數後遞歸。",
        "en": "If a terminal remainder greater than1 were composite,its least prime factor would already have been exhausted,a contradiction.The final entry is therefore prime.This certifies this instance;general existence follows by strong induction,recursively splitting a composite into smaller positive factors."
      },
      "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": "S372",
      "uses": [
        "G001",
        "S371"
      ],
      "before": "Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor462:input=462;COMPLETE prime multiset=[2, 3, 7, 11];distinct primes=[2, 3, 7, 11]",
      "after": "mu=1;empty product case=否;repeated prime=否",
      "relation": "implies",
      "rule_id": "mobius_value_one_repeated_prime_or_squarefree_parity",
      "conditions": [],
      "explanation": {
        "zh-Hant": "1的空prime多重集給mu1=1；任何prime重複表示square divisor而給0；否則依不同prime個數奇偶給正負。排序不丟重數。",
        "en": "The empty prime multiset for1 gives mu1=1.A repeated prime gives a square divisor and value0;otherwise distinct-prime parity determines the sign.Sorting must not discard multiplicities."
      },
      "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": "S373",
      "uses": [
        "G001",
        "S372"
      ],
      "before": "Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor462 := div(462, 462)",
      "after": "Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor462 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor462，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor462 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "462",
          "462"
        ],
        "output": "1",
        "label": "Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor462"
      }
    },
    {
      "id": "S374",
      "uses": [
        "G001",
        "S373"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/462):actual positive input=1;check complete integer range1..1",
      "after": "ALL positive divisors INCREASING=[1];not only primes/proper divisors or all integers<=n",
      "relation": "implies",
      "rule_id": "mobinv_ALL_positive_divisors_complete_indexing",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐完整範圍查整除，保1與n；d|n不是d<=n，也不只取prime或proper divisor。",
        "en": "Check divisibility over the complete range,retaining1 and n.The relation d|n is neither d<=n nor a prime/proper-divisor restriction."
      },
      "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": "S375",
      "uses": [
        "G001",
        "S374"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/462) f(1) ORIGINAL a times divisor := mul(20, 1)",
      "after": "Mobius inversion ORIGINAL inverse F(n/462) f(1) ORIGINAL a times divisor = 20",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/462) f(1) ORIGINAL a times divisor，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/462) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "20",
          "1"
        ],
        "output": "20",
        "label": "Mobius inversion ORIGINAL inverse F(n/462) f(1) ORIGINAL a times divisor"
      }
    },
    {
      "id": "S376",
      "uses": [
        "G001",
        "S375"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/462) f(1) plus ORIGINAL b := sum(20, 1)",
      "after": "Mobius inversion ORIGINAL inverse F(n/462) f(1) plus ORIGINAL b = 21",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/462) f(1) plus ORIGINAL b，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/462) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "20",
          "1"
        ],
        "output": "21",
        "label": "Mobius inversion ORIGINAL inverse F(n/462) f(1) plus ORIGINAL b"
      }
    },
    {
      "id": "S377",
      "uses": [
        "G001",
        "S376"
      ],
      "before": "Mobius inversion ORIGINAL inverse F(n/462) full forward divisor sum := sum(21)",
      "after": "Mobius inversion ORIGINAL inverse F(n/462) full forward divisor sum = 21",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse F(n/462) full forward divisor sum，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse F(n/462) full forward divisor sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "21"
        ],
        "output": "21",
        "label": "Mobius inversion ORIGINAL inverse F(n/462) full forward divisor sum"
      }
    },
    {
      "id": "S378",
      "uses": [
        "G001",
        "S377"
      ],
      "before": "Mobius inversion ORIGINAL inverse mu(462) times SAME F(n/462) := mul(1, 21)",
      "after": "Mobius inversion ORIGINAL inverse mu(462) times SAME F(n/462) = 21",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse mu(462) times SAME F(n/462)，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse mu(462) times SAME F(n/462) with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "21"
        ],
        "output": "21",
        "label": "Mobius inversion ORIGINAL inverse mu(462) times SAME F(n/462)"
      }
    },
    {
      "id": "S379",
      "uses": [
        "G001",
        "S378"
      ],
      "before": "Mobius inversion ORIGINAL inverse complete inverse-term sum := sum(23056, -7688, -5768, 1924, -2888, -1928, 964, 724, 644, 484, -242, -162, 244, -82, -62, 21)",
      "after": "Mobius inversion ORIGINAL inverse complete inverse-term sum = 9241",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse complete inverse-term sum，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse complete inverse-term sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "23056",
          "-7688",
          "-5768",
          "1924",
          "-2888",
          "-1928",
          "964",
          "724",
          "644",
          "484",
          "-242",
          "-162",
          "244",
          "-82",
          "-62",
          "21"
        ],
        "output": "9241",
        "label": "Mobius inversion ORIGINAL inverse complete inverse-term sum"
      }
    },
    {
      "id": "S380",
      "uses": [
        "G001",
        "S379"
      ],
      "before": "Mobius inversion ORIGINAL inverse direct ORIGINAL a times n := mul(20, 462)",
      "after": "Mobius inversion ORIGINAL inverse direct ORIGINAL a times n = 9240",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse direct ORIGINAL a times n，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse direct ORIGINAL a times n with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "20",
          "462"
        ],
        "output": "9240",
        "label": "Mobius inversion ORIGINAL inverse direct ORIGINAL a times n"
      }
    },
    {
      "id": "S381",
      "uses": [
        "G001",
        "S380"
      ],
      "before": "Mobius inversion ORIGINAL inverse direct plus ORIGINAL b := sum(9240, 1)",
      "after": "Mobius inversion ORIGINAL inverse direct plus ORIGINAL b = 9241",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL inverse direct plus ORIGINAL b，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL inverse direct plus ORIGINAL b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "9240",
          "1"
        ],
        "output": "9241",
        "label": "Mobius inversion ORIGINAL inverse direct plus ORIGINAL b"
      }
    },
    {
      "id": "S382",
      "uses": [
        "G001",
        "S381"
      ],
      "before": "Mobius inversion ORIGINAL inverse:actual n=462;ALL [d,mu(d),n/d,F(n/d),mu(d)*F(n/d)] rows=[[1, 1, 462, 23056, 23056], [2, -1, 231, 7688, -7688], [3, -1, 154, 5768, -5768], [6, 1, 77, 1924, 1924], [7, -1, 66, 2888, -2888], [11, -1, 42, 1928, -1928], [14, 1, 33, 964, 964], [21, 1, 22, 724, 724], [22, 1, 21, 644, 644], [33, 1, 14, 484, 484], [42, -1, 11, 242, -242], [66, -1, 7, 162, -162], [77, 1, 6, 244, 244], [154, -1, 3, 82, -82], [231, -1, 2, 62, -62], [462, 1, 1, 21, 21]]",
      "after": "complete reconstructed=9241;direct f(n)=9241;sum_d mu(d)sum_e|n/d f(e)=sum_e|n f(e)sum_d|n/e mu(d)=f(n)",
      "relation": "implies",
      "rule_id": "mobinv_general_finite_double_sum_and_Mobius_cancellation",
      "conditions": [],
      "explanation": {
        "zh-Hant": "有限雙和可換序，d*e|n；內層mu和在n/e=1時為1，否則由不同prime子集合正負配對為0，故只留e=n。n=1也有d=1、mu1=1，不混另一等價公式的index。",
        "en": "Swap the finite sums over d*e|n.The inner Mobius sum is1 for n/e=1 and0 otherwise by pairing distinct-prime subsets,leaving only e=n.For n1,the d1 term and mu1=1 remain;do not mix indices from the alternate equivalent convention."
      },
      "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": "S383",
      "uses": [
        "G001",
        "S382"
      ],
      "before": "Mobius inversion ORIGINAL omitted-divisor partial sum := sum(-7688, -5768, 1924, -2888, -1928, 964, 724, 644, 484, -242, -162, 244, -82, -62, 21)",
      "after": "Mobius inversion ORIGINAL omitted-divisor partial sum = -13815",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion ORIGINAL omitted-divisor partial sum，不以小數近似替代。",
          "en": "Recompute Mobius inversion ORIGINAL omitted-divisor partial sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-7688",
          "-5768",
          "1924",
          "-2888",
          "-1928",
          "964",
          "724",
          "644",
          "484",
          "-242",
          "-162",
          "244",
          "-82",
          "-62",
          "21"
        ],
        "output": "-13815",
        "label": "Mobius inversion ORIGINAL omitted-divisor partial sum"
      }
    },
    {
      "id": "S384",
      "uses": [
        "G001",
        "S383"
      ],
      "before": "Mobius inversion complete minus partial := sub(9241, -13815)",
      "after": "Mobius inversion complete minus partial = 23056",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。",
        "en": "Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Mobius inversion complete minus partial，不以小數近似替代。",
          "en": "Recompute Mobius inversion complete minus partial with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "9241",
          "-13815"
        ],
        "output": "23056",
        "label": "Mobius inversion complete minus partial"
      }
    },
    {
      "id": "S385",
      "uses": [
        "G001",
        "S384"
      ],
      "before": "ORIGINAL full divisors=[1, 2, 3, 6, 7, 11, 14, 21, 22, 33, 42, 66, 77, 154, 231, 462];actual omitted divisor=1;remaining ORIGINAL-order divisors=[2, 3, 6, 7, 11, 14, 21, 22, 33, 42, 66, 77, 154, 231, 462]",
      "after": "complete=9241=f(n)=9241;partial=-13815;lost term=23056!=0;incomplete_certificate fixed No",
      "relation": "implies",
      "rule_id": "mobinv_omission_actual_nonzero_discriminative_term",
      "conditions": [],
      "explanation": {
        "zh-Hant": "必須真漏一個divisor且其貢獻非零，使部分結果與目標不同；漏零項而恰巧相等不給本boundary證書。No拒絕不完整證書，不否認完整反演正確。",
        "en": "The omitted value must be an actual divisor with nonzero contribution,so the partial result differs from the target.Omitting a zero term does not qualify this discriminative boundary.The No rejects the incomplete certificate,not the complete inversion."
      },
      "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": [
      462,
      [
        1,
        2,
        3,
        6,
        7,
        11,
        14,
        21,
        22,
        33,
        42,
        66,
        77,
        154,
        231,
        462
      ],
      1,
      [
        2,
        3,
        6,
        7,
        11,
        14,
        21,
        22,
        33,
        42,
        66,
        77,
        154,
        231,
        462
      ],
      9241,
      -13815,
      9241,
      false,
      "all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required"
    ]
  },
  "conclusion": {
    "canonical_value": "(462, [1, 2, 3, 6, 7, 11, 14, 21, 22, 33, 42, 66, 77, 154, 231, 462], 1, [2, 3, 6, 7, 11, 14, 21, 22, 33, 42, 66, 77, 154, 231, 462], 9241, -13815, 9241, 否, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)",
    "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-20261110-r06-mobius-inversion-finite-divisors-foundations",
      "batch_path": "public/data/questions/batches/batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations/questions.jsonl",
      "batch_sha256": "3b7269ce09311ce689cb67967b7f9716e28457317a32efb8f14a079f0528cdf9",
      "original_line": 95,
      "original_line_sha256": "90d7c84acf2febeb6c6defa6fd2e14db4410655dff15b299331f5fe044f7dccf",
      "question_compact_sha256": "dddf2a7931dc250d7b88cb300d71e45113ffd09679958f4339c1fcfa7a826586",
      "english_row_compact_sha256": "c5905e60ff2dd7033288a6d9d425c4e35cc32f39e112993aa54615355e18b117",
      "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-mobius-inversion-finite-divisors": {
        "sha256": "903f02acdd799627c36103659ee206c7907a68eedbfb36d6a35a0e9dc6a2ef98",
        "english_sha256": "addf39b281bccdf34cabbe47f1f285b27a3040a3677596baafb9907ab773011d",
        "original_path": "public/data/mko/mobius-inversion-finite-divisors.json",
        "english_path": "public/data/i18n/editions/mko-mobius-inversion-finite-divisors.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "747b98e75085a6b55251b2d7e04bbf7c60cb5500e5047439b2798fb5f45538fa",
    "original_candidate_sha256": "fcbefe13998ceefac4663ca71c31e932cf64c51dbd089ec0d3546e8fc8a447a8",
    "original_candidate_line_sha256": "fcbefe13998ceefac4663ca71c31e932cf64c51dbd089ec0d3546e8fc8a447a8"
  },
  "historical_source_state": {
    "review_state": {
      "derivation": "candidate_authored_and_locally_checked",
      "source_question_state": "candidate",
      "independent_AI_or_human_review": "NOT_PERFORMED",
      "formalization": "not_formalized",
      "Lean": "NOT_RUN"
    },
    "scope_note": {
      "zh-Hant": "來源綁定教學推導候選；精確核對不等於獨立AI／人工審查或一般形式證明。原題、答案、MKO、公式、AST與歷史不變。",
      "en": "Source-bound pedagogical derivation candidate.Exact checking is not independent AI/human review or general formal proof.Original questions,answers,MKOs,formulas,ASTs and history remain unchanged."
    },
    "label": "Frozen producer metadata; current per-question review is separate below."
  },
  "revision": {
    "id": "review-20261026-009-p44",
    "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": "49cf6ec60211c92d1c0c302663af4d44479635dc8181fb61a20c3374288e21c5",
      "question_first_witness_sha256": "10fd25b4643e5ff12753daceaa61f9f2499bf381f65ef96e9d21185c7c0d0c5e"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 385,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "2d33dcc5b69d743eeba0f28f93dea7f579feb2b16eda09e74d7a4d9f1817818c",
      "arithmetic_steps": 264
    },
    "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."
  }
}