{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261013-advance-20261018-r04-prime-number-0100",
  "pilot_number": "R0017-09175",
  "number": 9175,
  "family_id": "qf-ext18-prime-interval-inventory",
  "title": {
    "zh-Hant": "第 9175 題：自然數質數與因數",
    "en": "Question 9175: Every Prime in an Inclusive Bounded Interval"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261013-advance-20261018-r04-prime-number-0100",
    "batch_id": "batch-20261013-advance-20261018-r04-prime-number-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-prime-number"
    ],
    "title_zh": "自然數質數與因數",
    "stem_zh": "自然數質數。操作=prime-interval-inventory；資料={\"lo\":120,\"hi\":140}。請依 [lo,hi,primes,count] 的欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(120, 140, [127, 131, 137, 139], 4)",
      "explanation_zh": "擴展基礎核對=(120, 140, [127, 131, 137, 139], 4)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；0與1不是質數；因數檢查必須完整到平方根界。"
    },
    "difficulty": {
      "band": "upper_secondary",
      "level": 3,
      "skills": [
        "自然數質數與因數"
      ]
    },
    "variant": {
      "family_id": "qf-ext18-prime-interval-inventory",
      "seed": 202610184100,
      "parameters": {
        "lo": 120,
        "hi": 140,
        "domain": "Nat_prime",
        "operation": "prime-interval-inventory",
        "semantic_template": "qf-ext18-prime-interval-inventory/{\"hi\":140,\"lo\":120}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_finite_arithmetic",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-13",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261013-advance-20261018-r04-prime-number-0100",
    "batch_id": "batch-20261013-advance-20261018-r04-prime-number-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "274c4926be047c9ef5686d8300e2349ae45e24f41d35c0454ab218f34130e6c6",
      "family_id": "qf-ext18-prime-interval-inventory",
      "seed": 202610184100,
      "semantic_template": "qf-ext18-prime-interval-inventory/{\"hi\":140,\"lo\":120}"
    },
    "fields": {
      "title": "Every Prime in an Inclusive Bounded Interval",
      "stem": "Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. lo=120, hi=140, with lo≤hi and a bounded window. Give (lo, hi, every prime in [lo,hi] in increasing order, count).",
      "answer": {
        "kind": "expression",
        "value": "(120, 140, [127, 131, 137, 139], 4)",
        "explanation": "Inspect every integer in the inclusive interval and retain 0/1 exclusions. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (120, 140, [127, 131, 137, 139], 4)."
      },
      "skills": [
        "Every Prime in an Inclusive Bounded Interval"
      ],
      "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": "自然數質數。操作=prime-interval-inventory；資料={\"lo\":120,\"hi\":140}。請依 [lo,hi,primes,count] 的欄序回答。",
        "en": "Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. lo=120, hi=140, with lo≤hi and a bounded window. Give (lo, hi, every prime in [lo,hi] in increasing order, count)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "natural-number prime protocol;0 and1 are not prime",
      "after": "a failed candidate witness alone is not a primality proof",
      "relation": "implies",
      "rule_id": "prime_complete_evidence_scope",
      "conditions": [],
      "explanation": {
        "zh-Hant": "自然數含0；質數判準須保留n≥2與完整因數界。單一因數候選失敗只否定該見證，不能直接斷言質數。",
        "en": "Naturals include0.Primality retains both n≥2 and a complete factor bound.A failed single factor candidate only invalidates that witness;it does not certify primality."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "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": "phase=interval value120;n=120",
      "after": "n>=2=是",
      "relation": "implies",
      "rule_id": "prime_lower_bound_guard",
      "conditions": [],
      "explanation": {
        "zh-Hant": "質數必須至少2；0與1先排除，不能把沒有非平凡試除因子誤當質數。",
        "en": "A prime must be at least2.Exclude0 and1 first rather than treating an absence of trial factors as primality."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "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": "interval value120 trial divisor2:natural dividend=120;positive divisor=2",
      "after": "120=60·2+0",
      "relation": "implies",
      "rule_id": "positive_euclidean_division",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數下完整列商餘等式，0≤r<divisor；不把取模0當可求值。",
        "en": "With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": {
        "operation": "euclidean_division",
        "inputs": [
          "120",
          "2",
          "60"
        ],
        "output": "0",
        "label": "interval value120 trial divisor2"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "phase=interval value120;tried ascending divisors=[2];sqrt floor=10",
      "after": "least factor at least2=2;prime=否",
      "relation": "implies",
      "rule_id": "complete_small_factor_bound",
      "conditions": [],
      "explanation": {
        "zh-Hant": "若n=uv合數且u,v≥2，至少一因子≤sqrt(n)。依序找到首個零餘數就是最小因子；完整界內皆無因子時n自己是最小至少2因子，故為質數。",
        "en": "If n=uv is composite with u,v≥2,one factor is≤sqrt(n).The first zero remainder in ascending order gives the least factor;if none exists through the full bound,n itself is the least factor≥2 and 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": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "phase=interval value121;n=121",
      "after": "n>=2=是",
      "relation": "implies",
      "rule_id": "prime_lower_bound_guard",
      "conditions": [],
      "explanation": {
        "zh-Hant": "質數必須至少2；0與1先排除，不能把沒有非平凡試除因子誤當質數。",
        "en": "A prime must be at least2.Exclude0 and1 first rather than treating an absence of trial factors as primality."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "interval value121 trial divisor2:natural dividend=121;positive divisor=2",
      "after": "121=60·2+1",
      "relation": "implies",
      "rule_id": "positive_euclidean_division",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數下完整列商餘等式，0≤r<divisor；不把取模0當可求值。",
        "en": "With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": {
        "operation": "euclidean_division",
        "inputs": [
          "121",
          "2",
          "60"
        ],
        "output": "1",
        "label": "interval value121 trial divisor2"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "interval value121 trial divisor3:natural dividend=121;positive divisor=3",
      "after": "121=40·3+1",
      "relation": "implies",
      "rule_id": "positive_euclidean_division",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數下完整列商餘等式，0≤r<divisor；不把取模0當可求值。",
        "en": "With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": {
        "operation": "euclidean_division",
        "inputs": [
          "121",
          "3",
          "40"
        ],
        "output": "1",
        "label": "interval value121 trial divisor3"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "interval value121 trial divisor4:natural dividend=121;positive divisor=4",
      "after": "121=30·4+1",
      "relation": "implies",
      "rule_id": "positive_euclidean_division",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數下完整列商餘等式，0≤r<divisor；不把取模0當可求值。",
        "en": "With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": {
        "operation": "euclidean_division",
        "inputs": [
          "121",
          "4",
          "30"
        ],
        "output": "1",
        "label": "interval value121 trial divisor4"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "interval value121 trial divisor5:natural dividend=121;positive divisor=5",
      "after": "121=24·5+1",
      "relation": "implies",
      "rule_id": "positive_euclidean_division",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數下完整列商餘等式，0≤r<divisor；不把取模0當可求值。",
        "en": "With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": {
        "operation": "euclidean_division",
        "inputs": [
          "121",
          "5",
          "24"
        ],
        "output": "1",
        "label": "interval value121 trial divisor5"
      }
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "interval value121 trial divisor6:natural dividend=121;positive divisor=6",
      "after": "121=20·6+1",
      "relation": "implies",
      "rule_id": "positive_euclidean_division",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數下完整列商餘等式，0≤r<divisor；不把取模0當可求值。",
        "en": "With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": {
        "operation": "euclidean_division",
        "inputs": [
          "121",
          "6",
          "20"
        ],
        "output": "1",
        "label": "interval value121 trial divisor6"
      }
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "interval value121 trial divisor7:natural dividend=121;positive divisor=7",
      "after": "121=17·7+2",
      "relation": "implies",
      "rule_id": "positive_euclidean_division",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數下完整列商餘等式，0≤r<divisor；不把取模0當可求值。",
        "en": "With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": {
        "operation": "euclidean_division",
        "inputs": [
          "121",
          "7",
          "17"
        ],
        "output": "2",
        "label": "interval value121 trial divisor7"
      }
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "interval value121 trial divisor8:natural dividend=121;positive divisor=8",
      "after": "121=15·8+1",
      "relation": "implies",
      "rule_id": "positive_euclidean_division",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數下完整列商餘等式，0≤r<divisor；不把取模0當可求值。",
        "en": "With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": {
        "operation": "euclidean_division",
        "inputs": [
          "121",
          "8",
          "15"
        ],
        "output": "1",
        "label": "interval value121 trial divisor8"
      }
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "interval value121 trial divisor9:natural dividend=121;positive divisor=9",
      "after": "121=13·9+4",
      "relation": "implies",
      "rule_id": "positive_euclidean_division",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數下完整列商餘等式，0≤r<divisor；不把取模0當可求值。",
        "en": "With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": {
        "operation": "euclidean_division",
        "inputs": [
          "121",
          "9",
          "13"
        ],
        "output": "4",
        "label": "interval value121 trial divisor9"
      }
    },
    {
      "id": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "interval value121 trial divisor10:natural dividend=121;positive divisor=10",
      "after": "121=12·10+1",
      "relation": "implies",
      "rule_id": "positive_euclidean_division",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數下完整列商餘等式，0≤r<divisor；不把取模0當可求值。",
        "en": "With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": {
        "operation": "euclidean_division",
        "inputs": [
          "121",
          "10",
          "12"
        ],
        "output": "1",
        "label": "interval value121 trial divisor10"
      }
    },
    {
      "id": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "interval value121 trial divisor11:natural dividend=121;positive divisor=11",
      "after": "121=11·11+0",
      "relation": "implies",
      "rule_id": "positive_euclidean_division",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數下完整列商餘等式，0≤r<divisor；不把取模0當可求值。",
        "en": "With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": {
        "operation": "euclidean_division",
        "inputs": [
          "121",
          "11",
          "11"
        ],
        "output": "0",
        "label": "interval value121 trial divisor11"
      }
    },
    {
      "id": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "phase=interval value121;tried ascending divisors=[2, 3, 4, 5, 6, 7, 8, 9, 10, 11];sqrt floor=11",
      "after": "least factor at least2=11;prime=否",
      "relation": "implies",
      "rule_id": "complete_small_factor_bound",
      "conditions": [],
      "explanation": {
        "zh-Hant": "若n=uv合數且u,v≥2，至少一因子≤sqrt(n)。依序找到首個零餘數就是最小因子；完整界內皆無因子時n自己是最小至少2因子，故為質數。",
        "en": "If n=uv is composite with u,v≥2,one factor is≤sqrt(n).The first zero remainder in ascending order gives the least factor;if none exists through the full bound,n itself is the least factor≥2 and 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": "S017",
      "uses": [
        "G001",
        "S016"
      ],
      "before": "phase=interval value122;n=122",
      "after": "n>=2=是",
      "relation": "implies",
      "rule_id": "prime_lower_bound_guard",
      "conditions": [],
      "explanation": {
        "zh-Hant": "質數必須至少2；0與1先排除，不能把沒有非平凡試除因子誤當質數。",
        "en": "A prime must be at least2.Exclude0 and1 first rather than treating an absence of trial factors as primality."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S018",
      "uses": [
        "G001",
        "S017"
      ],
      "before": "interval value122 trial divisor2:natural dividend=122;positive divisor=2",
      "after": "122=61·2+0",
      "relation": "implies",
      "rule_id": "positive_euclidean_division",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數下完整列商餘等式，0≤r<divisor；不把取模0當可求值。",
        "en": "With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": {
        "operation": "euclidean_division",
        "inputs": [
          "122",
          "2",
          "61"
        ],
        "output": "0",
        "label": "interval value122 trial divisor2"
      }
    },
    {
      "id": "S019",
      "uses": [
        "G001",
        "S018"
      ],
      "before": "phase=interval value122;tried ascending divisors=[2];sqrt floor=11",
      "after": "least factor at least2=2;prime=否",
      "relation": "implies",
      "rule_id": "complete_small_factor_bound",
      "conditions": [],
      "explanation": {
        "zh-Hant": "若n=uv合數且u,v≥2，至少一因子≤sqrt(n)。依序找到首個零餘數就是最小因子；完整界內皆無因子時n自己是最小至少2因子，故為質數。",
        "en": "If n=uv is composite with u,v≥2,one factor is≤sqrt(n).The first zero remainder in ascending order gives the least factor;if none exists through the full bound,n itself is the least factor≥2 and 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": "S020",
      "uses": [
        "G001",
        "S019"
      ],
      "before": "phase=interval value123;n=123",
      "after": "n>=2=是",
      "relation": "implies",
      "rule_id": "prime_lower_bound_guard",
      "conditions": [],
      "explanation": {
        "zh-Hant": "質數必須至少2；0與1先排除，不能把沒有非平凡試除因子誤當質數。",
        "en": "A prime must be at least2.Exclude0 and1 first rather than treating an absence of trial factors as primality."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S021",
      "uses": [
        "G001",
        "S020"
      ],
      "before": "interval value123 trial divisor2:natural dividend=123;positive divisor=2",
      "after": "123=61·2+1",
      "relation": "implies",
      "rule_id": "positive_euclidean_division",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數下完整列商餘等式，0≤r<divisor；不把取模0當可求值。",
        "en": "With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": {
        "operation": "euclidean_division",
        "inputs": [
          "123",
          "2",
          "61"
        ],
        "output": "1",
        "label": "interval value123 trial divisor2"
      }
    },
    {
      "id": "S022",
      "uses": [
        "G001",
        "S021"
      ],
      "before": "interval value123 trial divisor3:natural dividend=123;positive divisor=3",
      "after": "123=41·3+0",
      "relation": "implies",
      "rule_id": "positive_euclidean_division",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數下完整列商餘等式，0≤r<divisor；不把取模0當可求值。",
        "en": "With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": {
        "operation": "euclidean_division",
        "inputs": [
          "123",
          "3",
          "41"
        ],
        "output": "0",
        "label": "interval value123 trial divisor3"
      }
    },
    {
      "id": "S023",
      "uses": [
        "G001",
        "S022"
      ],
      "before": "phase=interval value123;tried ascending divisors=[2, 3];sqrt floor=11",
      "after": "least factor at least2=3;prime=否",
      "relation": "implies",
      "rule_id": "complete_small_factor_bound",
      "conditions": [],
      "explanation": {
        "zh-Hant": "若n=uv合數且u,v≥2，至少一因子≤sqrt(n)。依序找到首個零餘數就是最小因子；完整界內皆無因子時n自己是最小至少2因子，故為質數。",
        "en": "If n=uv is composite with u,v≥2,one factor is≤sqrt(n).The first zero remainder in ascending order gives the least factor;if none exists through the full bound,n itself is the least factor≥2 and 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": "S024",
      "uses": [
        "G001",
        "S023"
      ],
      "before": "phase=interval value124;n=124",
      "after": "n>=2=是",
      "relation": "implies",
      "rule_id": "prime_lower_bound_guard",
      "conditions": [],
      "explanation": {
        "zh-Hant": "質數必須至少2；0與1先排除，不能把沒有非平凡試除因子誤當質數。",
        "en": "A prime must be at least2.Exclude0 and1 first rather than treating an absence of trial factors as primality."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S025",
      "uses": [
        "G001",
        "S024"
      ],
      "before": "interval value124 trial divisor2:natural dividend=124;positive divisor=2",
      "after": "124=62·2+0",
      "relation": "implies",
      "rule_id": "positive_euclidean_division",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數下完整列商餘等式，0≤r<divisor；不把取模0當可求值。",
        "en": "With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": {
        "operation": "euclidean_division",
        "inputs": [
          "124",
          "2",
          "62"
        ],
        "output": "0",
        "label": "interval value124 trial divisor2"
      }
    },
    {
      "id": "S026",
      "uses": [
        "G001",
        "S025"
      ],
      "before": "phase=interval value124;tried ascending divisors=[2];sqrt floor=11",
      "after": "least factor at least2=2;prime=否",
      "relation": "implies",
      "rule_id": "complete_small_factor_bound",
      "conditions": [],
      "explanation": {
        "zh-Hant": "若n=uv合數且u,v≥2，至少一因子≤sqrt(n)。依序找到首個零餘數就是最小因子；完整界內皆無因子時n自己是最小至少2因子，故為質數。",
        "en": "If n=uv is composite with u,v≥2,one factor is≤sqrt(n).The first zero remainder in ascending order gives the least factor;if none exists through the full bound,n itself is the least factor≥2 and 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": "S027",
      "uses": [
        "G001",
        "S026"
      ],
      "before": "phase=interval value125;n=125",
      "after": "n>=2=是",
      "relation": "implies",
      "rule_id": "prime_lower_bound_guard",
      "conditions": [],
      "explanation": {
        "zh-Hant": "質數必須至少2；0與1先排除，不能把沒有非平凡試除因子誤當質數。",
        "en": "A prime must be at least2.Exclude0 and1 first rather than treating an absence of trial factors as primality."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S028",
      "uses": [
        "G001",
        "S027"
      ],
      "before": "interval value125 trial divisor2:natural dividend=125;positive divisor=2",
      "after": "125=62·2+1",
      "relation": "implies",
      "rule_id": "positive_euclidean_division",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數下完整列商餘等式，0≤r<divisor；不把取模0當可求值。",
        "en": "With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": {
        "operation": "euclidean_division",
        "inputs": [
          "125",
          "2",
          "62"
        ],
        "output": "1",
        "label": "interval value125 trial divisor2"
      }
    },
    {
      "id": "S029",
      "uses": [
        "G001",
        "S028"
      ],
      "before": "interval value125 trial divisor3:natural dividend=125;positive divisor=3",
      "after": "125=41·3+2",
      "relation": "implies",
      "rule_id": "positive_euclidean_division",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數下完整列商餘等式，0≤r<divisor；不把取模0當可求值。",
        "en": "With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": {
        "operation": "euclidean_division",
        "inputs": [
          "125",
          "3",
          "41"
        ],
        "output": "2",
        "label": "interval value125 trial divisor3"
      }
    },
    {
      "id": "S030",
      "uses": [
        "G001",
        "S029"
      ],
      "before": "interval value125 trial divisor4:natural dividend=125;positive divisor=4",
      "after": "125=31·4+1",
      "relation": "implies",
      "rule_id": "positive_euclidean_division",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數下完整列商餘等式，0≤r<divisor；不把取模0當可求值。",
        "en": "With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": {
        "operation": "euclidean_division",
        "inputs": [
          "125",
          "4",
          "31"
        ],
        "output": "1",
        "label": "interval value125 trial divisor4"
      }
    },
    {
      "id": "S031",
      "uses": [
        "G001",
        "S030"
      ],
      "before": "interval value125 trial divisor5:natural dividend=125;positive divisor=5",
      "after": "125=25·5+0",
      "relation": "implies",
      "rule_id": "positive_euclidean_division",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數下完整列商餘等式，0≤r<divisor；不把取模0當可求值。",
        "en": "With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": {
        "operation": "euclidean_division",
        "inputs": [
          "125",
          "5",
          "25"
        ],
        "output": "0",
        "label": "interval value125 trial divisor5"
      }
    },
    {
      "id": "S032",
      "uses": [
        "G001",
        "S031"
      ],
      "before": "phase=interval value125;tried ascending divisors=[2, 3, 4, 5];sqrt floor=11",
      "after": "least factor at least2=5;prime=否",
      "relation": "implies",
      "rule_id": "complete_small_factor_bound",
      "conditions": [],
      "explanation": {
        "zh-Hant": "若n=uv合數且u,v≥2，至少一因子≤sqrt(n)。依序找到首個零餘數就是最小因子；完整界內皆無因子時n自己是最小至少2因子，故為質數。",
        "en": "If n=uv is composite with u,v≥2,one factor is≤sqrt(n).The first zero remainder in ascending order gives the least factor;if none exists through the full bound,n itself is the least factor≥2 and 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": "S033",
      "uses": [
        "G001",
        "S032"
      ],
      "before": "phase=interval value126;n=126",
      "after": "n>=2=是",
      "relation": "implies",
      "rule_id": "prime_lower_bound_guard",
      "conditions": [],
      "explanation": {
        "zh-Hant": "質數必須至少2；0與1先排除，不能把沒有非平凡試除因子誤當質數。",
        "en": "A prime must be at least2.Exclude0 and1 first rather than treating an absence of trial factors as primality."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S034",
      "uses": [
        "G001",
        "S033"
      ],
      "before": "interval value126 trial divisor2:natural dividend=126;positive divisor=2",
      "after": "126=63·2+0",
      "relation": "implies",
      "rule_id": "positive_euclidean_division",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數下完整列商餘等式，0≤r<divisor；不把取模0當可求值。",
        "en": "With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": {
        "operation": "euclidean_division",
        "inputs": [
          "126",
          "2",
          "63"
        ],
        "output": "0",
        "label": "interval value126 trial divisor2"
      }
    },
    {
      "id": "S035",
      "uses": [
        "G001",
        "S034"
      ],
      "before": "phase=interval value126;tried ascending divisors=[2];sqrt floor=11",
      "after": "least factor at least2=2;prime=否",
      "relation": "implies",
      "rule_id": "complete_small_factor_bound",
      "conditions": [],
      "explanation": {
        "zh-Hant": "若n=uv合數且u,v≥2，至少一因子≤sqrt(n)。依序找到首個零餘數就是最小因子；完整界內皆無因子時n自己是最小至少2因子，故為質數。",
        "en": "If n=uv is composite with u,v≥2,one factor is≤sqrt(n).The first zero remainder in ascending order gives the least factor;if none exists through the full bound,n itself is the least factor≥2 and 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": "S036",
      "uses": [
        "G001",
        "S035"
      ],
      "before": "phase=interval value127;n=127",
      "after": "n>=2=是",
      "relation": "implies",
      "rule_id": "prime_lower_bound_guard",
      "conditions": [],
      "explanation": {
        "zh-Hant": "質數必須至少2；0與1先排除，不能把沒有非平凡試除因子誤當質數。",
        "en": "A prime must be at least2.Exclude0 and1 first rather than treating an absence of trial factors as primality."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S037",
      "uses": [
        "G001",
        "S036"
      ],
      "before": "interval value127 trial divisor2:natural dividend=127;positive divisor=2",
      "after": "127=63·2+1",
      "relation": "implies",
      "rule_id": "positive_euclidean_division",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數下完整列商餘等式，0≤r<divisor；不把取模0當可求值。",
        "en": "With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": {
        "operation": "euclidean_division",
        "inputs": [
          "127",
          "2",
          "63"
        ],
        "output": "1",
        "label": "interval value127 trial divisor2"
      }
    },
    {
      "id": "S038",
      "uses": [
        "G001",
        "S037"
      ],
      "before": "interval value127 trial divisor3:natural dividend=127;positive divisor=3",
      "after": "127=42·3+1",
      "relation": "implies",
      "rule_id": "positive_euclidean_division",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數下完整列商餘等式，0≤r<divisor；不把取模0當可求值。",
        "en": "With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": {
        "operation": "euclidean_division",
        "inputs": [
          "127",
          "3",
          "42"
        ],
        "output": "1",
        "label": "interval value127 trial divisor3"
      }
    },
    {
      "id": "S039",
      "uses": [
        "G001",
        "S038"
      ],
      "before": "interval value127 trial divisor4:natural dividend=127;positive divisor=4",
      "after": "127=31·4+3",
      "relation": "implies",
      "rule_id": "positive_euclidean_division",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數下完整列商餘等式，0≤r<divisor；不把取模0當可求值。",
        "en": "With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": {
        "operation": "euclidean_division",
        "inputs": [
          "127",
          "4",
          "31"
        ],
        "output": "3",
        "label": "interval value127 trial divisor4"
      }
    },
    {
      "id": "S040",
      "uses": [
        "G001",
        "S039"
      ],
      "before": "interval value127 trial divisor5:natural dividend=127;positive divisor=5",
      "after": "127=25·5+2",
      "relation": "implies",
      "rule_id": "positive_euclidean_division",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數下完整列商餘等式，0≤r<divisor；不把取模0當可求值。",
        "en": "With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": {
        "operation": "euclidean_division",
        "inputs": [
          "127",
          "5",
          "25"
        ],
        "output": "2",
        "label": "interval value127 trial divisor5"
      }
    },
    {
      "id": "S041",
      "uses": [
        "G001",
        "S040"
      ],
      "before": "interval value127 trial divisor6:natural dividend=127;positive divisor=6",
      "after": "127=21·6+1",
      "relation": "implies",
      "rule_id": "positive_euclidean_division",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數下完整列商餘等式，0≤r<divisor；不把取模0當可求值。",
        "en": "With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": {
        "operation": "euclidean_division",
        "inputs": [
          "127",
          "6",
          "21"
        ],
        "output": "1",
        "label": "interval value127 trial divisor6"
      }
    },
    {
      "id": "S042",
      "uses": [
        "G001",
        "S041"
      ],
      "before": "interval value127 trial divisor7:natural dividend=127;positive divisor=7",
      "after": "127=18·7+1",
      "relation": "implies",
      "rule_id": "positive_euclidean_division",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數下完整列商餘等式，0≤r<divisor；不把取模0當可求值。",
        "en": "With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": {
        "operation": "euclidean_division",
        "inputs": [
          "127",
          "7",
          "18"
        ],
        "output": "1",
        "label": "interval value127 trial divisor7"
      }
    },
    {
      "id": "S043",
      "uses": [
        "G001",
        "S042"
      ],
      "before": "interval value127 trial divisor8:natural dividend=127;positive divisor=8",
      "after": "127=15·8+7",
      "relation": "implies",
      "rule_id": "positive_euclidean_division",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數下完整列商餘等式，0≤r<divisor；不把取模0當可求值。",
        "en": "With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": {
        "operation": "euclidean_division",
        "inputs": [
          "127",
          "8",
          "15"
        ],
        "output": "7",
        "label": "interval value127 trial divisor8"
      }
    },
    {
      "id": "S044",
      "uses": [
        "G001",
        "S043"
      ],
      "before": "interval value127 trial divisor9:natural dividend=127;positive divisor=9",
      "after": "127=14·9+1",
      "relation": "implies",
      "rule_id": "positive_euclidean_division",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數下完整列商餘等式，0≤r<divisor；不把取模0當可求值。",
        "en": "With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": {
        "operation": "euclidean_division",
        "inputs": [
          "127",
          "9",
          "14"
        ],
        "output": "1",
        "label": "interval value127 trial divisor9"
      }
    },
    {
      "id": "S045",
      "uses": [
        "G001",
        "S044"
      ],
      "before": "interval value127 trial divisor10:natural dividend=127;positive divisor=10",
      "after": "127=12·10+7",
      "relation": "implies",
      "rule_id": "positive_euclidean_division",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數下完整列商餘等式，0≤r<divisor；不把取模0當可求值。",
        "en": "With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": {
        "operation": "euclidean_division",
        "inputs": [
          "127",
          "10",
          "12"
        ],
        "output": "7",
        "label": "interval value127 trial divisor10"
      }
    },
    {
      "id": "S046",
      "uses": [
        "G001",
        "S045"
      ],
      "before": "interval value127 trial divisor11:natural dividend=127;positive divisor=11",
      "after": "127=11·11+6",
      "relation": "implies",
      "rule_id": "positive_euclidean_division",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數下完整列商餘等式，0≤r<divisor；不把取模0當可求值。",
        "en": "With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": {
        "operation": "euclidean_division",
        "inputs": [
          "127",
          "11",
          "11"
        ],
        "output": "6",
        "label": "interval value127 trial divisor11"
      }
    },
    {
      "id": "S047",
      "uses": [
        "G001",
        "S046"
      ],
      "before": "phase=interval value127;tried ascending divisors=[2, 3, 4, 5, 6, 7, 8, 9, 10, 11];sqrt floor=11",
      "after": "least factor at least2=127;prime=是",
      "relation": "implies",
      "rule_id": "complete_small_factor_bound",
      "conditions": [],
      "explanation": {
        "zh-Hant": "若n=uv合數且u,v≥2，至少一因子≤sqrt(n)。依序找到首個零餘數就是最小因子；完整界內皆無因子時n自己是最小至少2因子，故為質數。",
        "en": "If n=uv is composite with u,v≥2,one factor is≤sqrt(n).The first zero remainder in ascending order gives the least factor;if none exists through the full bound,n itself is the least factor≥2 and 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": "S048",
      "uses": [
        "G001",
        "S047"
      ],
      "before": "phase=interval value128;n=128",
      "after": "n>=2=是",
      "relation": "implies",
      "rule_id": "prime_lower_bound_guard",
      "conditions": [],
      "explanation": {
        "zh-Hant": "質數必須至少2；0與1先排除，不能把沒有非平凡試除因子誤當質數。",
        "en": "A prime must be at least2.Exclude0 and1 first rather than treating an absence of trial factors as primality."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S049",
      "uses": [
        "G001",
        "S048"
      ],
      "before": "interval value128 trial divisor2:natural dividend=128;positive divisor=2",
      "after": "128=64·2+0",
      "relation": "implies",
      "rule_id": "positive_euclidean_division",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數下完整列商餘等式，0≤r<divisor；不把取模0當可求值。",
        "en": "With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": {
        "operation": "euclidean_division",
        "inputs": [
          "128",
          "2",
          "64"
        ],
        "output": "0",
        "label": "interval value128 trial divisor2"
      }
    },
    {
      "id": "S050",
      "uses": [
        "G001",
        "S049"
      ],
      "before": "phase=interval value128;tried ascending divisors=[2];sqrt floor=11",
      "after": "least factor at least2=2;prime=否",
      "relation": "implies",
      "rule_id": "complete_small_factor_bound",
      "conditions": [],
      "explanation": {
        "zh-Hant": "若n=uv合數且u,v≥2，至少一因子≤sqrt(n)。依序找到首個零餘數就是最小因子；完整界內皆無因子時n自己是最小至少2因子，故為質數。",
        "en": "If n=uv is composite with u,v≥2,one factor is≤sqrt(n).The first zero remainder in ascending order gives the least factor;if none exists through the full bound,n itself is the least factor≥2 and 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": "S051",
      "uses": [
        "G001",
        "S050"
      ],
      "before": "phase=interval value129;n=129",
      "after": "n>=2=是",
      "relation": "implies",
      "rule_id": "prime_lower_bound_guard",
      "conditions": [],
      "explanation": {
        "zh-Hant": "質數必須至少2；0與1先排除，不能把沒有非平凡試除因子誤當質數。",
        "en": "A prime must be at least2.Exclude0 and1 first rather than treating an absence of trial factors as primality."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S052",
      "uses": [
        "G001",
        "S051"
      ],
      "before": "interval value129 trial divisor2:natural dividend=129;positive divisor=2",
      "after": "129=64·2+1",
      "relation": "implies",
      "rule_id": "positive_euclidean_division",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數下完整列商餘等式，0≤r<divisor；不把取模0當可求值。",
        "en": "With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": {
        "operation": "euclidean_division",
        "inputs": [
          "129",
          "2",
          "64"
        ],
        "output": "1",
        "label": "interval value129 trial divisor2"
      }
    },
    {
      "id": "S053",
      "uses": [
        "G001",
        "S052"
      ],
      "before": "interval value129 trial divisor3:natural dividend=129;positive divisor=3",
      "after": "129=43·3+0",
      "relation": "implies",
      "rule_id": "positive_euclidean_division",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數下完整列商餘等式，0≤r<divisor；不把取模0當可求值。",
        "en": "With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": {
        "operation": "euclidean_division",
        "inputs": [
          "129",
          "3",
          "43"
        ],
        "output": "0",
        "label": "interval value129 trial divisor3"
      }
    },
    {
      "id": "S054",
      "uses": [
        "G001",
        "S053"
      ],
      "before": "phase=interval value129;tried ascending divisors=[2, 3];sqrt floor=11",
      "after": "least factor at least2=3;prime=否",
      "relation": "implies",
      "rule_id": "complete_small_factor_bound",
      "conditions": [],
      "explanation": {
        "zh-Hant": "若n=uv合數且u,v≥2，至少一因子≤sqrt(n)。依序找到首個零餘數就是最小因子；完整界內皆無因子時n自己是最小至少2因子，故為質數。",
        "en": "If n=uv is composite with u,v≥2,one factor is≤sqrt(n).The first zero remainder in ascending order gives the least factor;if none exists through the full bound,n itself is the least factor≥2 and 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": "S055",
      "uses": [
        "G001",
        "S054"
      ],
      "before": "phase=interval value130;n=130",
      "after": "n>=2=是",
      "relation": "implies",
      "rule_id": "prime_lower_bound_guard",
      "conditions": [],
      "explanation": {
        "zh-Hant": "質數必須至少2；0與1先排除，不能把沒有非平凡試除因子誤當質數。",
        "en": "A prime must be at least2.Exclude0 and1 first rather than treating an absence of trial factors as primality."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S056",
      "uses": [
        "G001",
        "S055"
      ],
      "before": "interval value130 trial divisor2:natural dividend=130;positive divisor=2",
      "after": "130=65·2+0",
      "relation": "implies",
      "rule_id": "positive_euclidean_division",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數下完整列商餘等式，0≤r<divisor；不把取模0當可求值。",
        "en": "With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": {
        "operation": "euclidean_division",
        "inputs": [
          "130",
          "2",
          "65"
        ],
        "output": "0",
        "label": "interval value130 trial divisor2"
      }
    },
    {
      "id": "S057",
      "uses": [
        "G001",
        "S056"
      ],
      "before": "phase=interval value130;tried ascending divisors=[2];sqrt floor=11",
      "after": "least factor at least2=2;prime=否",
      "relation": "implies",
      "rule_id": "complete_small_factor_bound",
      "conditions": [],
      "explanation": {
        "zh-Hant": "若n=uv合數且u,v≥2，至少一因子≤sqrt(n)。依序找到首個零餘數就是最小因子；完整界內皆無因子時n自己是最小至少2因子，故為質數。",
        "en": "If n=uv is composite with u,v≥2,one factor is≤sqrt(n).The first zero remainder in ascending order gives the least factor;if none exists through the full bound,n itself is the least factor≥2 and 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": "S058",
      "uses": [
        "G001",
        "S057"
      ],
      "before": "phase=interval value131;n=131",
      "after": "n>=2=是",
      "relation": "implies",
      "rule_id": "prime_lower_bound_guard",
      "conditions": [],
      "explanation": {
        "zh-Hant": "質數必須至少2；0與1先排除，不能把沒有非平凡試除因子誤當質數。",
        "en": "A prime must be at least2.Exclude0 and1 first rather than treating an absence of trial factors as primality."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S059",
      "uses": [
        "G001",
        "S058"
      ],
      "before": "interval value131 trial divisor2:natural dividend=131;positive divisor=2",
      "after": "131=65·2+1",
      "relation": "implies",
      "rule_id": "positive_euclidean_division",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數下完整列商餘等式，0≤r<divisor；不把取模0當可求值。",
        "en": "With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": {
        "operation": "euclidean_division",
        "inputs": [
          "131",
          "2",
          "65"
        ],
        "output": "1",
        "label": "interval value131 trial divisor2"
      }
    },
    {
      "id": "S060",
      "uses": [
        "G001",
        "S059"
      ],
      "before": "interval value131 trial divisor3:natural dividend=131;positive divisor=3",
      "after": "131=43·3+2",
      "relation": "implies",
      "rule_id": "positive_euclidean_division",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數下完整列商餘等式，0≤r<divisor；不把取模0當可求值。",
        "en": "With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": {
        "operation": "euclidean_division",
        "inputs": [
          "131",
          "3",
          "43"
        ],
        "output": "2",
        "label": "interval value131 trial divisor3"
      }
    },
    {
      "id": "S061",
      "uses": [
        "G001",
        "S060"
      ],
      "before": "interval value131 trial divisor4:natural dividend=131;positive divisor=4",
      "after": "131=32·4+3",
      "relation": "implies",
      "rule_id": "positive_euclidean_division",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數下完整列商餘等式，0≤r<divisor；不把取模0當可求值。",
        "en": "With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": {
        "operation": "euclidean_division",
        "inputs": [
          "131",
          "4",
          "32"
        ],
        "output": "3",
        "label": "interval value131 trial divisor4"
      }
    },
    {
      "id": "S062",
      "uses": [
        "G001",
        "S061"
      ],
      "before": "interval value131 trial divisor5:natural dividend=131;positive divisor=5",
      "after": "131=26·5+1",
      "relation": "implies",
      "rule_id": "positive_euclidean_division",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數下完整列商餘等式，0≤r<divisor；不把取模0當可求值。",
        "en": "With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": {
        "operation": "euclidean_division",
        "inputs": [
          "131",
          "5",
          "26"
        ],
        "output": "1",
        "label": "interval value131 trial divisor5"
      }
    },
    {
      "id": "S063",
      "uses": [
        "G001",
        "S062"
      ],
      "before": "interval value131 trial divisor6:natural dividend=131;positive divisor=6",
      "after": "131=21·6+5",
      "relation": "implies",
      "rule_id": "positive_euclidean_division",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數下完整列商餘等式，0≤r<divisor；不把取模0當可求值。",
        "en": "With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": {
        "operation": "euclidean_division",
        "inputs": [
          "131",
          "6",
          "21"
        ],
        "output": "5",
        "label": "interval value131 trial divisor6"
      }
    },
    {
      "id": "S064",
      "uses": [
        "G001",
        "S063"
      ],
      "before": "interval value131 trial divisor7:natural dividend=131;positive divisor=7",
      "after": "131=18·7+5",
      "relation": "implies",
      "rule_id": "positive_euclidean_division",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數下完整列商餘等式，0≤r<divisor；不把取模0當可求值。",
        "en": "With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": {
        "operation": "euclidean_division",
        "inputs": [
          "131",
          "7",
          "18"
        ],
        "output": "5",
        "label": "interval value131 trial divisor7"
      }
    },
    {
      "id": "S065",
      "uses": [
        "G001",
        "S064"
      ],
      "before": "interval value131 trial divisor8:natural dividend=131;positive divisor=8",
      "after": "131=16·8+3",
      "relation": "implies",
      "rule_id": "positive_euclidean_division",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數下完整列商餘等式，0≤r<divisor；不把取模0當可求值。",
        "en": "With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": {
        "operation": "euclidean_division",
        "inputs": [
          "131",
          "8",
          "16"
        ],
        "output": "3",
        "label": "interval value131 trial divisor8"
      }
    },
    {
      "id": "S066",
      "uses": [
        "G001",
        "S065"
      ],
      "before": "interval value131 trial divisor9:natural dividend=131;positive divisor=9",
      "after": "131=14·9+5",
      "relation": "implies",
      "rule_id": "positive_euclidean_division",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數下完整列商餘等式，0≤r<divisor；不把取模0當可求值。",
        "en": "With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": {
        "operation": "euclidean_division",
        "inputs": [
          "131",
          "9",
          "14"
        ],
        "output": "5",
        "label": "interval value131 trial divisor9"
      }
    },
    {
      "id": "S067",
      "uses": [
        "G001",
        "S066"
      ],
      "before": "interval value131 trial divisor10:natural dividend=131;positive divisor=10",
      "after": "131=13·10+1",
      "relation": "implies",
      "rule_id": "positive_euclidean_division",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數下完整列商餘等式，0≤r<divisor；不把取模0當可求值。",
        "en": "With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": {
        "operation": "euclidean_division",
        "inputs": [
          "131",
          "10",
          "13"
        ],
        "output": "1",
        "label": "interval value131 trial divisor10"
      }
    },
    {
      "id": "S068",
      "uses": [
        "G001",
        "S067"
      ],
      "before": "interval value131 trial divisor11:natural dividend=131;positive divisor=11",
      "after": "131=11·11+10",
      "relation": "implies",
      "rule_id": "positive_euclidean_division",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數下完整列商餘等式，0≤r<divisor；不把取模0當可求值。",
        "en": "With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": {
        "operation": "euclidean_division",
        "inputs": [
          "131",
          "11",
          "11"
        ],
        "output": "10",
        "label": "interval value131 trial divisor11"
      }
    },
    {
      "id": "S069",
      "uses": [
        "G001",
        "S068"
      ],
      "before": "phase=interval value131;tried ascending divisors=[2, 3, 4, 5, 6, 7, 8, 9, 10, 11];sqrt floor=11",
      "after": "least factor at least2=131;prime=是",
      "relation": "implies",
      "rule_id": "complete_small_factor_bound",
      "conditions": [],
      "explanation": {
        "zh-Hant": "若n=uv合數且u,v≥2，至少一因子≤sqrt(n)。依序找到首個零餘數就是最小因子；完整界內皆無因子時n自己是最小至少2因子，故為質數。",
        "en": "If n=uv is composite with u,v≥2,one factor is≤sqrt(n).The first zero remainder in ascending order gives the least factor;if none exists through the full bound,n itself is the least factor≥2 and 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": "S070",
      "uses": [
        "G001",
        "S069"
      ],
      "before": "phase=interval value132;n=132",
      "after": "n>=2=是",
      "relation": "implies",
      "rule_id": "prime_lower_bound_guard",
      "conditions": [],
      "explanation": {
        "zh-Hant": "質數必須至少2；0與1先排除，不能把沒有非平凡試除因子誤當質數。",
        "en": "A prime must be at least2.Exclude0 and1 first rather than treating an absence of trial factors as primality."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "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": "interval value132 trial divisor2:natural dividend=132;positive divisor=2",
      "after": "132=66·2+0",
      "relation": "implies",
      "rule_id": "positive_euclidean_division",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數下完整列商餘等式，0≤r<divisor；不把取模0當可求值。",
        "en": "With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": {
        "operation": "euclidean_division",
        "inputs": [
          "132",
          "2",
          "66"
        ],
        "output": "0",
        "label": "interval value132 trial divisor2"
      }
    },
    {
      "id": "S072",
      "uses": [
        "G001",
        "S071"
      ],
      "before": "phase=interval value132;tried ascending divisors=[2];sqrt floor=11",
      "after": "least factor at least2=2;prime=否",
      "relation": "implies",
      "rule_id": "complete_small_factor_bound",
      "conditions": [],
      "explanation": {
        "zh-Hant": "若n=uv合數且u,v≥2，至少一因子≤sqrt(n)。依序找到首個零餘數就是最小因子；完整界內皆無因子時n自己是最小至少2因子，故為質數。",
        "en": "If n=uv is composite with u,v≥2,one factor is≤sqrt(n).The first zero remainder in ascending order gives the least factor;if none exists through the full bound,n itself is the least factor≥2 and 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": "S073",
      "uses": [
        "G001",
        "S072"
      ],
      "before": "phase=interval value133;n=133",
      "after": "n>=2=是",
      "relation": "implies",
      "rule_id": "prime_lower_bound_guard",
      "conditions": [],
      "explanation": {
        "zh-Hant": "質數必須至少2；0與1先排除，不能把沒有非平凡試除因子誤當質數。",
        "en": "A prime must be at least2.Exclude0 and1 first rather than treating an absence of trial factors as primality."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S074",
      "uses": [
        "G001",
        "S073"
      ],
      "before": "interval value133 trial divisor2:natural dividend=133;positive divisor=2",
      "after": "133=66·2+1",
      "relation": "implies",
      "rule_id": "positive_euclidean_division",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數下完整列商餘等式，0≤r<divisor；不把取模0當可求值。",
        "en": "With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": {
        "operation": "euclidean_division",
        "inputs": [
          "133",
          "2",
          "66"
        ],
        "output": "1",
        "label": "interval value133 trial divisor2"
      }
    },
    {
      "id": "S075",
      "uses": [
        "G001",
        "S074"
      ],
      "before": "interval value133 trial divisor3:natural dividend=133;positive divisor=3",
      "after": "133=44·3+1",
      "relation": "implies",
      "rule_id": "positive_euclidean_division",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數下完整列商餘等式，0≤r<divisor；不把取模0當可求值。",
        "en": "With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": {
        "operation": "euclidean_division",
        "inputs": [
          "133",
          "3",
          "44"
        ],
        "output": "1",
        "label": "interval value133 trial divisor3"
      }
    },
    {
      "id": "S076",
      "uses": [
        "G001",
        "S075"
      ],
      "before": "interval value133 trial divisor4:natural dividend=133;positive divisor=4",
      "after": "133=33·4+1",
      "relation": "implies",
      "rule_id": "positive_euclidean_division",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數下完整列商餘等式，0≤r<divisor；不把取模0當可求值。",
        "en": "With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": {
        "operation": "euclidean_division",
        "inputs": [
          "133",
          "4",
          "33"
        ],
        "output": "1",
        "label": "interval value133 trial divisor4"
      }
    },
    {
      "id": "S077",
      "uses": [
        "G001",
        "S076"
      ],
      "before": "interval value133 trial divisor5:natural dividend=133;positive divisor=5",
      "after": "133=26·5+3",
      "relation": "implies",
      "rule_id": "positive_euclidean_division",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數下完整列商餘等式，0≤r<divisor；不把取模0當可求值。",
        "en": "With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": {
        "operation": "euclidean_division",
        "inputs": [
          "133",
          "5",
          "26"
        ],
        "output": "3",
        "label": "interval value133 trial divisor5"
      }
    },
    {
      "id": "S078",
      "uses": [
        "G001",
        "S077"
      ],
      "before": "interval value133 trial divisor6:natural dividend=133;positive divisor=6",
      "after": "133=22·6+1",
      "relation": "implies",
      "rule_id": "positive_euclidean_division",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數下完整列商餘等式，0≤r<divisor；不把取模0當可求值。",
        "en": "With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": {
        "operation": "euclidean_division",
        "inputs": [
          "133",
          "6",
          "22"
        ],
        "output": "1",
        "label": "interval value133 trial divisor6"
      }
    },
    {
      "id": "S079",
      "uses": [
        "G001",
        "S078"
      ],
      "before": "interval value133 trial divisor7:natural dividend=133;positive divisor=7",
      "after": "133=19·7+0",
      "relation": "implies",
      "rule_id": "positive_euclidean_division",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數下完整列商餘等式，0≤r<divisor；不把取模0當可求值。",
        "en": "With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": {
        "operation": "euclidean_division",
        "inputs": [
          "133",
          "7",
          "19"
        ],
        "output": "0",
        "label": "interval value133 trial divisor7"
      }
    },
    {
      "id": "S080",
      "uses": [
        "G001",
        "S079"
      ],
      "before": "phase=interval value133;tried ascending divisors=[2, 3, 4, 5, 6, 7];sqrt floor=11",
      "after": "least factor at least2=7;prime=否",
      "relation": "implies",
      "rule_id": "complete_small_factor_bound",
      "conditions": [],
      "explanation": {
        "zh-Hant": "若n=uv合數且u,v≥2，至少一因子≤sqrt(n)。依序找到首個零餘數就是最小因子；完整界內皆無因子時n自己是最小至少2因子，故為質數。",
        "en": "If n=uv is composite with u,v≥2,one factor is≤sqrt(n).The first zero remainder in ascending order gives the least factor;if none exists through the full bound,n itself is the least factor≥2 and 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": "S081",
      "uses": [
        "G001",
        "S080"
      ],
      "before": "phase=interval value134;n=134",
      "after": "n>=2=是",
      "relation": "implies",
      "rule_id": "prime_lower_bound_guard",
      "conditions": [],
      "explanation": {
        "zh-Hant": "質數必須至少2；0與1先排除，不能把沒有非平凡試除因子誤當質數。",
        "en": "A prime must be at least2.Exclude0 and1 first rather than treating an absence of trial factors as primality."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S082",
      "uses": [
        "G001",
        "S081"
      ],
      "before": "interval value134 trial divisor2:natural dividend=134;positive divisor=2",
      "after": "134=67·2+0",
      "relation": "implies",
      "rule_id": "positive_euclidean_division",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數下完整列商餘等式，0≤r<divisor；不把取模0當可求值。",
        "en": "With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": {
        "operation": "euclidean_division",
        "inputs": [
          "134",
          "2",
          "67"
        ],
        "output": "0",
        "label": "interval value134 trial divisor2"
      }
    },
    {
      "id": "S083",
      "uses": [
        "G001",
        "S082"
      ],
      "before": "phase=interval value134;tried ascending divisors=[2];sqrt floor=11",
      "after": "least factor at least2=2;prime=否",
      "relation": "implies",
      "rule_id": "complete_small_factor_bound",
      "conditions": [],
      "explanation": {
        "zh-Hant": "若n=uv合數且u,v≥2，至少一因子≤sqrt(n)。依序找到首個零餘數就是最小因子；完整界內皆無因子時n自己是最小至少2因子，故為質數。",
        "en": "If n=uv is composite with u,v≥2,one factor is≤sqrt(n).The first zero remainder in ascending order gives the least factor;if none exists through the full bound,n itself is the least factor≥2 and 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": "S084",
      "uses": [
        "G001",
        "S083"
      ],
      "before": "phase=interval value135;n=135",
      "after": "n>=2=是",
      "relation": "implies",
      "rule_id": "prime_lower_bound_guard",
      "conditions": [],
      "explanation": {
        "zh-Hant": "質數必須至少2；0與1先排除，不能把沒有非平凡試除因子誤當質數。",
        "en": "A prime must be at least2.Exclude0 and1 first rather than treating an absence of trial factors as primality."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S085",
      "uses": [
        "G001",
        "S084"
      ],
      "before": "interval value135 trial divisor2:natural dividend=135;positive divisor=2",
      "after": "135=67·2+1",
      "relation": "implies",
      "rule_id": "positive_euclidean_division",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數下完整列商餘等式，0≤r<divisor；不把取模0當可求值。",
        "en": "With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": {
        "operation": "euclidean_division",
        "inputs": [
          "135",
          "2",
          "67"
        ],
        "output": "1",
        "label": "interval value135 trial divisor2"
      }
    },
    {
      "id": "S086",
      "uses": [
        "G001",
        "S085"
      ],
      "before": "interval value135 trial divisor3:natural dividend=135;positive divisor=3",
      "after": "135=45·3+0",
      "relation": "implies",
      "rule_id": "positive_euclidean_division",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數下完整列商餘等式，0≤r<divisor；不把取模0當可求值。",
        "en": "With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": {
        "operation": "euclidean_division",
        "inputs": [
          "135",
          "3",
          "45"
        ],
        "output": "0",
        "label": "interval value135 trial divisor3"
      }
    },
    {
      "id": "S087",
      "uses": [
        "G001",
        "S086"
      ],
      "before": "phase=interval value135;tried ascending divisors=[2, 3];sqrt floor=11",
      "after": "least factor at least2=3;prime=否",
      "relation": "implies",
      "rule_id": "complete_small_factor_bound",
      "conditions": [],
      "explanation": {
        "zh-Hant": "若n=uv合數且u,v≥2，至少一因子≤sqrt(n)。依序找到首個零餘數就是最小因子；完整界內皆無因子時n自己是最小至少2因子，故為質數。",
        "en": "If n=uv is composite with u,v≥2,one factor is≤sqrt(n).The first zero remainder in ascending order gives the least factor;if none exists through the full bound,n itself is the least factor≥2 and 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": "S088",
      "uses": [
        "G001",
        "S087"
      ],
      "before": "phase=interval value136;n=136",
      "after": "n>=2=是",
      "relation": "implies",
      "rule_id": "prime_lower_bound_guard",
      "conditions": [],
      "explanation": {
        "zh-Hant": "質數必須至少2；0與1先排除，不能把沒有非平凡試除因子誤當質數。",
        "en": "A prime must be at least2.Exclude0 and1 first rather than treating an absence of trial factors as primality."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S089",
      "uses": [
        "G001",
        "S088"
      ],
      "before": "interval value136 trial divisor2:natural dividend=136;positive divisor=2",
      "after": "136=68·2+0",
      "relation": "implies",
      "rule_id": "positive_euclidean_division",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數下完整列商餘等式，0≤r<divisor；不把取模0當可求值。",
        "en": "With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": {
        "operation": "euclidean_division",
        "inputs": [
          "136",
          "2",
          "68"
        ],
        "output": "0",
        "label": "interval value136 trial divisor2"
      }
    },
    {
      "id": "S090",
      "uses": [
        "G001",
        "S089"
      ],
      "before": "phase=interval value136;tried ascending divisors=[2];sqrt floor=11",
      "after": "least factor at least2=2;prime=否",
      "relation": "implies",
      "rule_id": "complete_small_factor_bound",
      "conditions": [],
      "explanation": {
        "zh-Hant": "若n=uv合數且u,v≥2，至少一因子≤sqrt(n)。依序找到首個零餘數就是最小因子；完整界內皆無因子時n自己是最小至少2因子，故為質數。",
        "en": "If n=uv is composite with u,v≥2,one factor is≤sqrt(n).The first zero remainder in ascending order gives the least factor;if none exists through the full bound,n itself is the least factor≥2 and 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": "S091",
      "uses": [
        "G001",
        "S090"
      ],
      "before": "phase=interval value137;n=137",
      "after": "n>=2=是",
      "relation": "implies",
      "rule_id": "prime_lower_bound_guard",
      "conditions": [],
      "explanation": {
        "zh-Hant": "質數必須至少2；0與1先排除，不能把沒有非平凡試除因子誤當質數。",
        "en": "A prime must be at least2.Exclude0 and1 first rather than treating an absence of trial factors as primality."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "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": "interval value137 trial divisor2:natural dividend=137;positive divisor=2",
      "after": "137=68·2+1",
      "relation": "implies",
      "rule_id": "positive_euclidean_division",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數下完整列商餘等式，0≤r<divisor；不把取模0當可求值。",
        "en": "With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": {
        "operation": "euclidean_division",
        "inputs": [
          "137",
          "2",
          "68"
        ],
        "output": "1",
        "label": "interval value137 trial divisor2"
      }
    },
    {
      "id": "S093",
      "uses": [
        "G001",
        "S092"
      ],
      "before": "interval value137 trial divisor3:natural dividend=137;positive divisor=3",
      "after": "137=45·3+2",
      "relation": "implies",
      "rule_id": "positive_euclidean_division",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數下完整列商餘等式，0≤r<divisor；不把取模0當可求值。",
        "en": "With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": {
        "operation": "euclidean_division",
        "inputs": [
          "137",
          "3",
          "45"
        ],
        "output": "2",
        "label": "interval value137 trial divisor3"
      }
    },
    {
      "id": "S094",
      "uses": [
        "G001",
        "S093"
      ],
      "before": "interval value137 trial divisor4:natural dividend=137;positive divisor=4",
      "after": "137=34·4+1",
      "relation": "implies",
      "rule_id": "positive_euclidean_division",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數下完整列商餘等式，0≤r<divisor；不把取模0當可求值。",
        "en": "With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": {
        "operation": "euclidean_division",
        "inputs": [
          "137",
          "4",
          "34"
        ],
        "output": "1",
        "label": "interval value137 trial divisor4"
      }
    },
    {
      "id": "S095",
      "uses": [
        "G001",
        "S094"
      ],
      "before": "interval value137 trial divisor5:natural dividend=137;positive divisor=5",
      "after": "137=27·5+2",
      "relation": "implies",
      "rule_id": "positive_euclidean_division",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數下完整列商餘等式，0≤r<divisor；不把取模0當可求值。",
        "en": "With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": {
        "operation": "euclidean_division",
        "inputs": [
          "137",
          "5",
          "27"
        ],
        "output": "2",
        "label": "interval value137 trial divisor5"
      }
    },
    {
      "id": "S096",
      "uses": [
        "G001",
        "S095"
      ],
      "before": "interval value137 trial divisor6:natural dividend=137;positive divisor=6",
      "after": "137=22·6+5",
      "relation": "implies",
      "rule_id": "positive_euclidean_division",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數下完整列商餘等式，0≤r<divisor；不把取模0當可求值。",
        "en": "With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": {
        "operation": "euclidean_division",
        "inputs": [
          "137",
          "6",
          "22"
        ],
        "output": "5",
        "label": "interval value137 trial divisor6"
      }
    },
    {
      "id": "S097",
      "uses": [
        "G001",
        "S096"
      ],
      "before": "interval value137 trial divisor7:natural dividend=137;positive divisor=7",
      "after": "137=19·7+4",
      "relation": "implies",
      "rule_id": "positive_euclidean_division",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數下完整列商餘等式，0≤r<divisor；不把取模0當可求值。",
        "en": "With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": {
        "operation": "euclidean_division",
        "inputs": [
          "137",
          "7",
          "19"
        ],
        "output": "4",
        "label": "interval value137 trial divisor7"
      }
    },
    {
      "id": "S098",
      "uses": [
        "G001",
        "S097"
      ],
      "before": "interval value137 trial divisor8:natural dividend=137;positive divisor=8",
      "after": "137=17·8+1",
      "relation": "implies",
      "rule_id": "positive_euclidean_division",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數下完整列商餘等式，0≤r<divisor；不把取模0當可求值。",
        "en": "With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": {
        "operation": "euclidean_division",
        "inputs": [
          "137",
          "8",
          "17"
        ],
        "output": "1",
        "label": "interval value137 trial divisor8"
      }
    },
    {
      "id": "S099",
      "uses": [
        "G001",
        "S098"
      ],
      "before": "interval value137 trial divisor9:natural dividend=137;positive divisor=9",
      "after": "137=15·9+2",
      "relation": "implies",
      "rule_id": "positive_euclidean_division",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數下完整列商餘等式，0≤r<divisor；不把取模0當可求值。",
        "en": "With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": {
        "operation": "euclidean_division",
        "inputs": [
          "137",
          "9",
          "15"
        ],
        "output": "2",
        "label": "interval value137 trial divisor9"
      }
    },
    {
      "id": "S100",
      "uses": [
        "G001",
        "S099"
      ],
      "before": "interval value137 trial divisor10:natural dividend=137;positive divisor=10",
      "after": "137=13·10+7",
      "relation": "implies",
      "rule_id": "positive_euclidean_division",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數下完整列商餘等式，0≤r<divisor；不把取模0當可求值。",
        "en": "With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": {
        "operation": "euclidean_division",
        "inputs": [
          "137",
          "10",
          "13"
        ],
        "output": "7",
        "label": "interval value137 trial divisor10"
      }
    },
    {
      "id": "S101",
      "uses": [
        "G001",
        "S100"
      ],
      "before": "interval value137 trial divisor11:natural dividend=137;positive divisor=11",
      "after": "137=12·11+5",
      "relation": "implies",
      "rule_id": "positive_euclidean_division",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數下完整列商餘等式，0≤r<divisor；不把取模0當可求值。",
        "en": "With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": {
        "operation": "euclidean_division",
        "inputs": [
          "137",
          "11",
          "12"
        ],
        "output": "5",
        "label": "interval value137 trial divisor11"
      }
    },
    {
      "id": "S102",
      "uses": [
        "G001",
        "S101"
      ],
      "before": "phase=interval value137;tried ascending divisors=[2, 3, 4, 5, 6, 7, 8, 9, 10, 11];sqrt floor=11",
      "after": "least factor at least2=137;prime=是",
      "relation": "implies",
      "rule_id": "complete_small_factor_bound",
      "conditions": [],
      "explanation": {
        "zh-Hant": "若n=uv合數且u,v≥2，至少一因子≤sqrt(n)。依序找到首個零餘數就是最小因子；完整界內皆無因子時n自己是最小至少2因子，故為質數。",
        "en": "If n=uv is composite with u,v≥2,one factor is≤sqrt(n).The first zero remainder in ascending order gives the least factor;if none exists through the full bound,n itself is the least factor≥2 and 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": "S103",
      "uses": [
        "G001",
        "S102"
      ],
      "before": "phase=interval value138;n=138",
      "after": "n>=2=是",
      "relation": "implies",
      "rule_id": "prime_lower_bound_guard",
      "conditions": [],
      "explanation": {
        "zh-Hant": "質數必須至少2；0與1先排除，不能把沒有非平凡試除因子誤當質數。",
        "en": "A prime must be at least2.Exclude0 and1 first rather than treating an absence of trial factors as primality."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S104",
      "uses": [
        "G001",
        "S103"
      ],
      "before": "interval value138 trial divisor2:natural dividend=138;positive divisor=2",
      "after": "138=69·2+0",
      "relation": "implies",
      "rule_id": "positive_euclidean_division",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數下完整列商餘等式，0≤r<divisor；不把取模0當可求值。",
        "en": "With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": {
        "operation": "euclidean_division",
        "inputs": [
          "138",
          "2",
          "69"
        ],
        "output": "0",
        "label": "interval value138 trial divisor2"
      }
    },
    {
      "id": "S105",
      "uses": [
        "G001",
        "S104"
      ],
      "before": "phase=interval value138;tried ascending divisors=[2];sqrt floor=11",
      "after": "least factor at least2=2;prime=否",
      "relation": "implies",
      "rule_id": "complete_small_factor_bound",
      "conditions": [],
      "explanation": {
        "zh-Hant": "若n=uv合數且u,v≥2，至少一因子≤sqrt(n)。依序找到首個零餘數就是最小因子；完整界內皆無因子時n自己是最小至少2因子，故為質數。",
        "en": "If n=uv is composite with u,v≥2,one factor is≤sqrt(n).The first zero remainder in ascending order gives the least factor;if none exists through the full bound,n itself is the least factor≥2 and 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": "S106",
      "uses": [
        "G001",
        "S105"
      ],
      "before": "phase=interval value139;n=139",
      "after": "n>=2=是",
      "relation": "implies",
      "rule_id": "prime_lower_bound_guard",
      "conditions": [],
      "explanation": {
        "zh-Hant": "質數必須至少2；0與1先排除，不能把沒有非平凡試除因子誤當質數。",
        "en": "A prime must be at least2.Exclude0 and1 first rather than treating an absence of trial factors as primality."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S107",
      "uses": [
        "G001",
        "S106"
      ],
      "before": "interval value139 trial divisor2:natural dividend=139;positive divisor=2",
      "after": "139=69·2+1",
      "relation": "implies",
      "rule_id": "positive_euclidean_division",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數下完整列商餘等式，0≤r<divisor；不把取模0當可求值。",
        "en": "With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": {
        "operation": "euclidean_division",
        "inputs": [
          "139",
          "2",
          "69"
        ],
        "output": "1",
        "label": "interval value139 trial divisor2"
      }
    },
    {
      "id": "S108",
      "uses": [
        "G001",
        "S107"
      ],
      "before": "interval value139 trial divisor3:natural dividend=139;positive divisor=3",
      "after": "139=46·3+1",
      "relation": "implies",
      "rule_id": "positive_euclidean_division",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數下完整列商餘等式，0≤r<divisor；不把取模0當可求值。",
        "en": "With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": {
        "operation": "euclidean_division",
        "inputs": [
          "139",
          "3",
          "46"
        ],
        "output": "1",
        "label": "interval value139 trial divisor3"
      }
    },
    {
      "id": "S109",
      "uses": [
        "G001",
        "S108"
      ],
      "before": "interval value139 trial divisor4:natural dividend=139;positive divisor=4",
      "after": "139=34·4+3",
      "relation": "implies",
      "rule_id": "positive_euclidean_division",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數下完整列商餘等式，0≤r<divisor；不把取模0當可求值。",
        "en": "With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": {
        "operation": "euclidean_division",
        "inputs": [
          "139",
          "4",
          "34"
        ],
        "output": "3",
        "label": "interval value139 trial divisor4"
      }
    },
    {
      "id": "S110",
      "uses": [
        "G001",
        "S109"
      ],
      "before": "interval value139 trial divisor5:natural dividend=139;positive divisor=5",
      "after": "139=27·5+4",
      "relation": "implies",
      "rule_id": "positive_euclidean_division",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數下完整列商餘等式，0≤r<divisor；不把取模0當可求值。",
        "en": "With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": {
        "operation": "euclidean_division",
        "inputs": [
          "139",
          "5",
          "27"
        ],
        "output": "4",
        "label": "interval value139 trial divisor5"
      }
    },
    {
      "id": "S111",
      "uses": [
        "G001",
        "S110"
      ],
      "before": "interval value139 trial divisor6:natural dividend=139;positive divisor=6",
      "after": "139=23·6+1",
      "relation": "implies",
      "rule_id": "positive_euclidean_division",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數下完整列商餘等式，0≤r<divisor；不把取模0當可求值。",
        "en": "With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": {
        "operation": "euclidean_division",
        "inputs": [
          "139",
          "6",
          "23"
        ],
        "output": "1",
        "label": "interval value139 trial divisor6"
      }
    },
    {
      "id": "S112",
      "uses": [
        "G001",
        "S111"
      ],
      "before": "interval value139 trial divisor7:natural dividend=139;positive divisor=7",
      "after": "139=19·7+6",
      "relation": "implies",
      "rule_id": "positive_euclidean_division",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數下完整列商餘等式，0≤r<divisor；不把取模0當可求值。",
        "en": "With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": {
        "operation": "euclidean_division",
        "inputs": [
          "139",
          "7",
          "19"
        ],
        "output": "6",
        "label": "interval value139 trial divisor7"
      }
    },
    {
      "id": "S113",
      "uses": [
        "G001",
        "S112"
      ],
      "before": "interval value139 trial divisor8:natural dividend=139;positive divisor=8",
      "after": "139=17·8+3",
      "relation": "implies",
      "rule_id": "positive_euclidean_division",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數下完整列商餘等式，0≤r<divisor；不把取模0當可求值。",
        "en": "With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": {
        "operation": "euclidean_division",
        "inputs": [
          "139",
          "8",
          "17"
        ],
        "output": "3",
        "label": "interval value139 trial divisor8"
      }
    },
    {
      "id": "S114",
      "uses": [
        "G001",
        "S113"
      ],
      "before": "interval value139 trial divisor9:natural dividend=139;positive divisor=9",
      "after": "139=15·9+4",
      "relation": "implies",
      "rule_id": "positive_euclidean_division",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數下完整列商餘等式，0≤r<divisor；不把取模0當可求值。",
        "en": "With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": {
        "operation": "euclidean_division",
        "inputs": [
          "139",
          "9",
          "15"
        ],
        "output": "4",
        "label": "interval value139 trial divisor9"
      }
    },
    {
      "id": "S115",
      "uses": [
        "G001",
        "S114"
      ],
      "before": "interval value139 trial divisor10:natural dividend=139;positive divisor=10",
      "after": "139=13·10+9",
      "relation": "implies",
      "rule_id": "positive_euclidean_division",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數下完整列商餘等式，0≤r<divisor；不把取模0當可求值。",
        "en": "With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": {
        "operation": "euclidean_division",
        "inputs": [
          "139",
          "10",
          "13"
        ],
        "output": "9",
        "label": "interval value139 trial divisor10"
      }
    },
    {
      "id": "S116",
      "uses": [
        "G001",
        "S115"
      ],
      "before": "interval value139 trial divisor11:natural dividend=139;positive divisor=11",
      "after": "139=12·11+7",
      "relation": "implies",
      "rule_id": "positive_euclidean_division",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數下完整列商餘等式，0≤r<divisor；不把取模0當可求值。",
        "en": "With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": {
        "operation": "euclidean_division",
        "inputs": [
          "139",
          "11",
          "12"
        ],
        "output": "7",
        "label": "interval value139 trial divisor11"
      }
    },
    {
      "id": "S117",
      "uses": [
        "G001",
        "S116"
      ],
      "before": "phase=interval value139;tried ascending divisors=[2, 3, 4, 5, 6, 7, 8, 9, 10, 11];sqrt floor=11",
      "after": "least factor at least2=139;prime=是",
      "relation": "implies",
      "rule_id": "complete_small_factor_bound",
      "conditions": [],
      "explanation": {
        "zh-Hant": "若n=uv合數且u,v≥2，至少一因子≤sqrt(n)。依序找到首個零餘數就是最小因子；完整界內皆無因子時n自己是最小至少2因子，故為質數。",
        "en": "If n=uv is composite with u,v≥2,one factor is≤sqrt(n).The first zero remainder in ascending order gives the least factor;if none exists through the full bound,n itself is the least factor≥2 and 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": "S118",
      "uses": [
        "G001",
        "S117"
      ],
      "before": "phase=interval value140;n=140",
      "after": "n>=2=是",
      "relation": "implies",
      "rule_id": "prime_lower_bound_guard",
      "conditions": [],
      "explanation": {
        "zh-Hant": "質數必須至少2；0與1先排除，不能把沒有非平凡試除因子誤當質數。",
        "en": "A prime must be at least2.Exclude0 and1 first rather than treating an absence of trial factors as primality."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "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": "interval value140 trial divisor2:natural dividend=140;positive divisor=2",
      "after": "140=70·2+0",
      "relation": "implies",
      "rule_id": "positive_euclidean_division",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數下完整列商餘等式，0≤r<divisor；不把取模0當可求值。",
        "en": "With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": {
        "operation": "euclidean_division",
        "inputs": [
          "140",
          "2",
          "70"
        ],
        "output": "0",
        "label": "interval value140 trial divisor2"
      }
    },
    {
      "id": "S120",
      "uses": [
        "G001",
        "S119"
      ],
      "before": "phase=interval value140;tried ascending divisors=[2];sqrt floor=11",
      "after": "least factor at least2=2;prime=否",
      "relation": "implies",
      "rule_id": "complete_small_factor_bound",
      "conditions": [],
      "explanation": {
        "zh-Hant": "若n=uv合數且u,v≥2，至少一因子≤sqrt(n)。依序找到首個零餘數就是最小因子；完整界內皆無因子時n自己是最小至少2因子，故為質數。",
        "en": "If n=uv is composite with u,v≥2,one factor is≤sqrt(n).The first zero remainder in ascending order gives the least factor;if none exists through the full bound,n itself is the least factor≥2 and 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": "S121",
      "uses": [
        "G001",
        "S120"
      ],
      "before": "inclusive interval=[120, 140]",
      "after": "all primes=[127, 131, 137, 139];count=4",
      "relation": "implies",
      "rule_id": "complete_inclusive_prime_interval",
      "conditions": [],
      "explanation": {
        "zh-Hant": "區間兩端都核，逐一保留且只保留通過完整質數判準的數；沒有以有限列舉推論區間外的一般分布。",
        "en": "Check both interval endpoints and retain exactly those values passing the complete primality test.This finite enumeration makes no distribution claim outside the interval."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "result": [
      120,
      140,
      [
        127,
        131,
        137,
        139
      ],
      4
    ]
  },
  "conclusion": {
    "canonical_value": "(120, 140, [127, 131, 137, 139], 4)",
    "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-20261013-advance-20261018-r04-prime-number-foundations",
      "batch_path": "public/data/questions/batches/batch-20261013-advance-20261018-r04-prime-number-foundations/questions.jsonl",
      "batch_sha256": "39b1f12c357ca8c69d6a9660a72667958bd462bae4faf3ab07254018c9cfafe4",
      "original_line": 100,
      "original_line_sha256": "8759fb3acbe103252afcbe1909ee8b9e79289b819223ec73a8d90affa876c28d",
      "question_compact_sha256": "274c4926be047c9ef5686d8300e2349ae45e24f41d35c0454ab218f34130e6c6",
      "english_row_compact_sha256": "951331fe74638a74d3318e92a0565cec2cc4eae457cfdb2513110f10dece0cd7",
      "english_manifest_path": "public/data/i18n/editions/batch-20261013-advance-20261018-r04-prime-number-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261013-advance-20261018-r04-prime-number-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-prime-number": {
        "sha256": "5ebeb1330ba52c9685111730d0c0c2255e653887f8869f119ebbbb767ed139b5",
        "english_sha256": "9c7980d6fc0591adf0ccc93fefbab485b5b58c30d23d24e43095b2a8f18fd5aa",
        "original_path": "public/data/mko/prime-number.json",
        "english_path": "public/data/i18n/editions/mko-prime-number.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "20e59f30c0b6218e6d9614c16170d770e56a2f1e4ae5ce39d0a8a3db206afa61",
    "original_candidate_sha256": "244f19890a8221ef817e7c9409c5ce17caa91cdeda7793aeea43b23677df6f7f",
    "original_candidate_line_sha256": "244f19890a8221ef817e7c9409c5ce17caa91cdeda7793aeea43b23677df6f7f"
  },
  "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-20261026-006-p18",
    "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": "abad28564830bb9b6ef069ca5a1aa128ce7b1a67a5cb8fe6d868a06db73bcfb8",
      "question_first_witness_sha256": "43126988479968837abe086263dac1b24c37463ad80e00d2e280a7b8e9036a04"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 121,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "c587ed92579d97084b2e24a88835f3bd080ed4cb2f7929e6b093bc1279c756b9",
      "arithmetic_steps": 77
    },
    "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."
  }
}