{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261013-advance-20261018-r04-prime-number-0093",
  "pilot_number": "R0017-09168",
  "number": 9168,
  "family_id": "qf-ext18-prime-interval-inventory",
  "title": {
    "zh-Hant": "第 9168 題：自然數質數與因數",
    "en": "Question 9168: Every Prime in an Inclusive Bounded Interval"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261013-advance-20261018-r04-prime-number-0093",
    "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\":85,\"hi\":105}。請依 [lo,hi,primes,count] 的欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(85, 105, [89, 97, 101, 103], 4)",
      "explanation_zh": "擴展基礎核對=(85, 105, [89, 97, 101, 103], 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": 202610184093,
      "parameters": {
        "lo": 85,
        "hi": 105,
        "domain": "Nat_prime",
        "operation": "prime-interval-inventory",
        "semantic_template": "qf-ext18-prime-interval-inventory/{\"hi\":105,\"lo\":85}"
      }
    },
    "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-0093",
    "batch_id": "batch-20261013-advance-20261018-r04-prime-number-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "2318c0290f9c140adec69313a93596042ddebdfaa6fc213cc745667064cd7708",
      "family_id": "qf-ext18-prime-interval-inventory",
      "seed": 202610184093,
      "semantic_template": "qf-ext18-prime-interval-inventory/{\"hi\":105,\"lo\":85}"
    },
    "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=85, hi=105, with lo≤hi and a bounded window. Give (lo, hi, every prime in [lo,hi] in increasing order, count).",
      "answer": {
        "kind": "expression",
        "value": "(85, 105, [89, 97, 101, 103], 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 (85, 105, [89, 97, 101, 103], 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\":85,\"hi\":105}。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=85, hi=105, 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 value85;n=85",
      "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 value85 trial divisor2:natural dividend=85;positive divisor=2",
      "after": "85=42·2+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": [
          "85",
          "2",
          "42"
        ],
        "output": "1",
        "label": "interval value85 trial divisor2"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "interval value85 trial divisor3:natural dividend=85;positive divisor=3",
      "after": "85=28·3+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": [
          "85",
          "3",
          "28"
        ],
        "output": "1",
        "label": "interval value85 trial divisor3"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "interval value85 trial divisor4:natural dividend=85;positive divisor=4",
      "after": "85=21·4+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": [
          "85",
          "4",
          "21"
        ],
        "output": "1",
        "label": "interval value85 trial divisor4"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "interval value85 trial divisor5:natural dividend=85;positive divisor=5",
      "after": "85=17·5+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": [
          "85",
          "5",
          "17"
        ],
        "output": "0",
        "label": "interval value85 trial divisor5"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "phase=interval value85;tried ascending divisors=[2, 3, 4, 5];sqrt floor=9",
      "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": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "phase=interval value86;n=86",
      "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": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "interval value86 trial divisor2:natural dividend=86;positive divisor=2",
      "after": "86=43·2+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": [
          "86",
          "2",
          "43"
        ],
        "output": "0",
        "label": "interval value86 trial divisor2"
      }
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "phase=interval value86;tried ascending divisors=[2];sqrt floor=9",
      "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": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "phase=interval value87;n=87",
      "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": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "interval value87 trial divisor2:natural dividend=87;positive divisor=2",
      "after": "87=43·2+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": [
          "87",
          "2",
          "43"
        ],
        "output": "1",
        "label": "interval value87 trial divisor2"
      }
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "interval value87 trial divisor3:natural dividend=87;positive divisor=3",
      "after": "87=29·3+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": [
          "87",
          "3",
          "29"
        ],
        "output": "0",
        "label": "interval value87 trial divisor3"
      }
    },
    {
      "id": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "phase=interval value87;tried ascending divisors=[2, 3];sqrt floor=9",
      "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": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "phase=interval value88;n=88",
      "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": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "interval value88 trial divisor2:natural dividend=88;positive divisor=2",
      "after": "88=44·2+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": [
          "88",
          "2",
          "44"
        ],
        "output": "0",
        "label": "interval value88 trial divisor2"
      }
    },
    {
      "id": "S017",
      "uses": [
        "G001",
        "S016"
      ],
      "before": "phase=interval value88;tried ascending divisors=[2];sqrt floor=9",
      "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": "S018",
      "uses": [
        "G001",
        "S017"
      ],
      "before": "phase=interval value89;n=89",
      "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": "S019",
      "uses": [
        "G001",
        "S018"
      ],
      "before": "interval value89 trial divisor2:natural dividend=89;positive divisor=2",
      "after": "89=44·2+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": [
          "89",
          "2",
          "44"
        ],
        "output": "1",
        "label": "interval value89 trial divisor2"
      }
    },
    {
      "id": "S020",
      "uses": [
        "G001",
        "S019"
      ],
      "before": "interval value89 trial divisor3:natural dividend=89;positive divisor=3",
      "after": "89=29·3+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": [
          "89",
          "3",
          "29"
        ],
        "output": "2",
        "label": "interval value89 trial divisor3"
      }
    },
    {
      "id": "S021",
      "uses": [
        "G001",
        "S020"
      ],
      "before": "interval value89 trial divisor4:natural dividend=89;positive divisor=4",
      "after": "89=22·4+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": [
          "89",
          "4",
          "22"
        ],
        "output": "1",
        "label": "interval value89 trial divisor4"
      }
    },
    {
      "id": "S022",
      "uses": [
        "G001",
        "S021"
      ],
      "before": "interval value89 trial divisor5:natural dividend=89;positive divisor=5",
      "after": "89=17·5+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": [
          "89",
          "5",
          "17"
        ],
        "output": "4",
        "label": "interval value89 trial divisor5"
      }
    },
    {
      "id": "S023",
      "uses": [
        "G001",
        "S022"
      ],
      "before": "interval value89 trial divisor6:natural dividend=89;positive divisor=6",
      "after": "89=14·6+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": [
          "89",
          "6",
          "14"
        ],
        "output": "5",
        "label": "interval value89 trial divisor6"
      }
    },
    {
      "id": "S024",
      "uses": [
        "G001",
        "S023"
      ],
      "before": "interval value89 trial divisor7:natural dividend=89;positive divisor=7",
      "after": "89=12·7+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": [
          "89",
          "7",
          "12"
        ],
        "output": "5",
        "label": "interval value89 trial divisor7"
      }
    },
    {
      "id": "S025",
      "uses": [
        "G001",
        "S024"
      ],
      "before": "interval value89 trial divisor8:natural dividend=89;positive divisor=8",
      "after": "89=11·8+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": [
          "89",
          "8",
          "11"
        ],
        "output": "1",
        "label": "interval value89 trial divisor8"
      }
    },
    {
      "id": "S026",
      "uses": [
        "G001",
        "S025"
      ],
      "before": "interval value89 trial divisor9:natural dividend=89;positive divisor=9",
      "after": "89=9·9+8",
      "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": [
          "89",
          "9",
          "9"
        ],
        "output": "8",
        "label": "interval value89 trial divisor9"
      }
    },
    {
      "id": "S027",
      "uses": [
        "G001",
        "S026"
      ],
      "before": "phase=interval value89;tried ascending divisors=[2, 3, 4, 5, 6, 7, 8, 9];sqrt floor=9",
      "after": "least factor at least2=89;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": "S028",
      "uses": [
        "G001",
        "S027"
      ],
      "before": "phase=interval value90;n=90",
      "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": "S029",
      "uses": [
        "G001",
        "S028"
      ],
      "before": "interval value90 trial divisor2:natural dividend=90;positive divisor=2",
      "after": "90=45·2+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": [
          "90",
          "2",
          "45"
        ],
        "output": "0",
        "label": "interval value90 trial divisor2"
      }
    },
    {
      "id": "S030",
      "uses": [
        "G001",
        "S029"
      ],
      "before": "phase=interval value90;tried ascending divisors=[2];sqrt floor=9",
      "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": "S031",
      "uses": [
        "G001",
        "S030"
      ],
      "before": "phase=interval value91;n=91",
      "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": "S032",
      "uses": [
        "G001",
        "S031"
      ],
      "before": "interval value91 trial divisor2:natural dividend=91;positive divisor=2",
      "after": "91=45·2+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": [
          "91",
          "2",
          "45"
        ],
        "output": "1",
        "label": "interval value91 trial divisor2"
      }
    },
    {
      "id": "S033",
      "uses": [
        "G001",
        "S032"
      ],
      "before": "interval value91 trial divisor3:natural dividend=91;positive divisor=3",
      "after": "91=30·3+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": [
          "91",
          "3",
          "30"
        ],
        "output": "1",
        "label": "interval value91 trial divisor3"
      }
    },
    {
      "id": "S034",
      "uses": [
        "G001",
        "S033"
      ],
      "before": "interval value91 trial divisor4:natural dividend=91;positive divisor=4",
      "after": "91=22·4+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": [
          "91",
          "4",
          "22"
        ],
        "output": "3",
        "label": "interval value91 trial divisor4"
      }
    },
    {
      "id": "S035",
      "uses": [
        "G001",
        "S034"
      ],
      "before": "interval value91 trial divisor5:natural dividend=91;positive divisor=5",
      "after": "91=18·5+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": [
          "91",
          "5",
          "18"
        ],
        "output": "1",
        "label": "interval value91 trial divisor5"
      }
    },
    {
      "id": "S036",
      "uses": [
        "G001",
        "S035"
      ],
      "before": "interval value91 trial divisor6:natural dividend=91;positive divisor=6",
      "after": "91=15·6+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": [
          "91",
          "6",
          "15"
        ],
        "output": "1",
        "label": "interval value91 trial divisor6"
      }
    },
    {
      "id": "S037",
      "uses": [
        "G001",
        "S036"
      ],
      "before": "interval value91 trial divisor7:natural dividend=91;positive divisor=7",
      "after": "91=13·7+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": [
          "91",
          "7",
          "13"
        ],
        "output": "0",
        "label": "interval value91 trial divisor7"
      }
    },
    {
      "id": "S038",
      "uses": [
        "G001",
        "S037"
      ],
      "before": "phase=interval value91;tried ascending divisors=[2, 3, 4, 5, 6, 7];sqrt floor=9",
      "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": "S039",
      "uses": [
        "G001",
        "S038"
      ],
      "before": "phase=interval value92;n=92",
      "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": "S040",
      "uses": [
        "G001",
        "S039"
      ],
      "before": "interval value92 trial divisor2:natural dividend=92;positive divisor=2",
      "after": "92=46·2+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": [
          "92",
          "2",
          "46"
        ],
        "output": "0",
        "label": "interval value92 trial divisor2"
      }
    },
    {
      "id": "S041",
      "uses": [
        "G001",
        "S040"
      ],
      "before": "phase=interval value92;tried ascending divisors=[2];sqrt floor=9",
      "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": "S042",
      "uses": [
        "G001",
        "S041"
      ],
      "before": "phase=interval value93;n=93",
      "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": "S043",
      "uses": [
        "G001",
        "S042"
      ],
      "before": "interval value93 trial divisor2:natural dividend=93;positive divisor=2",
      "after": "93=46·2+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": [
          "93",
          "2",
          "46"
        ],
        "output": "1",
        "label": "interval value93 trial divisor2"
      }
    },
    {
      "id": "S044",
      "uses": [
        "G001",
        "S043"
      ],
      "before": "interval value93 trial divisor3:natural dividend=93;positive divisor=3",
      "after": "93=31·3+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": [
          "93",
          "3",
          "31"
        ],
        "output": "0",
        "label": "interval value93 trial divisor3"
      }
    },
    {
      "id": "S045",
      "uses": [
        "G001",
        "S044"
      ],
      "before": "phase=interval value93;tried ascending divisors=[2, 3];sqrt floor=9",
      "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": "S046",
      "uses": [
        "G001",
        "S045"
      ],
      "before": "phase=interval value94;n=94",
      "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": "S047",
      "uses": [
        "G001",
        "S046"
      ],
      "before": "interval value94 trial divisor2:natural dividend=94;positive divisor=2",
      "after": "94=47·2+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": [
          "94",
          "2",
          "47"
        ],
        "output": "0",
        "label": "interval value94 trial divisor2"
      }
    },
    {
      "id": "S048",
      "uses": [
        "G001",
        "S047"
      ],
      "before": "phase=interval value94;tried ascending divisors=[2];sqrt floor=9",
      "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": "S049",
      "uses": [
        "G001",
        "S048"
      ],
      "before": "phase=interval value95;n=95",
      "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": "S050",
      "uses": [
        "G001",
        "S049"
      ],
      "before": "interval value95 trial divisor2:natural dividend=95;positive divisor=2",
      "after": "95=47·2+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": [
          "95",
          "2",
          "47"
        ],
        "output": "1",
        "label": "interval value95 trial divisor2"
      }
    },
    {
      "id": "S051",
      "uses": [
        "G001",
        "S050"
      ],
      "before": "interval value95 trial divisor3:natural dividend=95;positive divisor=3",
      "after": "95=31·3+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": [
          "95",
          "3",
          "31"
        ],
        "output": "2",
        "label": "interval value95 trial divisor3"
      }
    },
    {
      "id": "S052",
      "uses": [
        "G001",
        "S051"
      ],
      "before": "interval value95 trial divisor4:natural dividend=95;positive divisor=4",
      "after": "95=23·4+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": [
          "95",
          "4",
          "23"
        ],
        "output": "3",
        "label": "interval value95 trial divisor4"
      }
    },
    {
      "id": "S053",
      "uses": [
        "G001",
        "S052"
      ],
      "before": "interval value95 trial divisor5:natural dividend=95;positive divisor=5",
      "after": "95=19·5+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": [
          "95",
          "5",
          "19"
        ],
        "output": "0",
        "label": "interval value95 trial divisor5"
      }
    },
    {
      "id": "S054",
      "uses": [
        "G001",
        "S053"
      ],
      "before": "phase=interval value95;tried ascending divisors=[2, 3, 4, 5];sqrt floor=9",
      "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": "S055",
      "uses": [
        "G001",
        "S054"
      ],
      "before": "phase=interval value96;n=96",
      "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 value96 trial divisor2:natural dividend=96;positive divisor=2",
      "after": "96=48·2+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": [
          "96",
          "2",
          "48"
        ],
        "output": "0",
        "label": "interval value96 trial divisor2"
      }
    },
    {
      "id": "S057",
      "uses": [
        "G001",
        "S056"
      ],
      "before": "phase=interval value96;tried ascending divisors=[2];sqrt floor=9",
      "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 value97;n=97",
      "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 value97 trial divisor2:natural dividend=97;positive divisor=2",
      "after": "97=48·2+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": [
          "97",
          "2",
          "48"
        ],
        "output": "1",
        "label": "interval value97 trial divisor2"
      }
    },
    {
      "id": "S060",
      "uses": [
        "G001",
        "S059"
      ],
      "before": "interval value97 trial divisor3:natural dividend=97;positive divisor=3",
      "after": "97=32·3+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": [
          "97",
          "3",
          "32"
        ],
        "output": "1",
        "label": "interval value97 trial divisor3"
      }
    },
    {
      "id": "S061",
      "uses": [
        "G001",
        "S060"
      ],
      "before": "interval value97 trial divisor4:natural dividend=97;positive divisor=4",
      "after": "97=24·4+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": [
          "97",
          "4",
          "24"
        ],
        "output": "1",
        "label": "interval value97 trial divisor4"
      }
    },
    {
      "id": "S062",
      "uses": [
        "G001",
        "S061"
      ],
      "before": "interval value97 trial divisor5:natural dividend=97;positive divisor=5",
      "after": "97=19·5+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": [
          "97",
          "5",
          "19"
        ],
        "output": "2",
        "label": "interval value97 trial divisor5"
      }
    },
    {
      "id": "S063",
      "uses": [
        "G001",
        "S062"
      ],
      "before": "interval value97 trial divisor6:natural dividend=97;positive divisor=6",
      "after": "97=16·6+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": [
          "97",
          "6",
          "16"
        ],
        "output": "1",
        "label": "interval value97 trial divisor6"
      }
    },
    {
      "id": "S064",
      "uses": [
        "G001",
        "S063"
      ],
      "before": "interval value97 trial divisor7:natural dividend=97;positive divisor=7",
      "after": "97=13·7+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": [
          "97",
          "7",
          "13"
        ],
        "output": "6",
        "label": "interval value97 trial divisor7"
      }
    },
    {
      "id": "S065",
      "uses": [
        "G001",
        "S064"
      ],
      "before": "interval value97 trial divisor8:natural dividend=97;positive divisor=8",
      "after": "97=12·8+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": [
          "97",
          "8",
          "12"
        ],
        "output": "1",
        "label": "interval value97 trial divisor8"
      }
    },
    {
      "id": "S066",
      "uses": [
        "G001",
        "S065"
      ],
      "before": "interval value97 trial divisor9:natural dividend=97;positive divisor=9",
      "after": "97=10·9+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": [
          "97",
          "9",
          "10"
        ],
        "output": "7",
        "label": "interval value97 trial divisor9"
      }
    },
    {
      "id": "S067",
      "uses": [
        "G001",
        "S066"
      ],
      "before": "phase=interval value97;tried ascending divisors=[2, 3, 4, 5, 6, 7, 8, 9];sqrt floor=9",
      "after": "least factor at least2=97;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": "S068",
      "uses": [
        "G001",
        "S067"
      ],
      "before": "phase=interval value98;n=98",
      "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": "S069",
      "uses": [
        "G001",
        "S068"
      ],
      "before": "interval value98 trial divisor2:natural dividend=98;positive divisor=2",
      "after": "98=49·2+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": [
          "98",
          "2",
          "49"
        ],
        "output": "0",
        "label": "interval value98 trial divisor2"
      }
    },
    {
      "id": "S070",
      "uses": [
        "G001",
        "S069"
      ],
      "before": "phase=interval value98;tried ascending divisors=[2];sqrt floor=9",
      "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": "S071",
      "uses": [
        "G001",
        "S070"
      ],
      "before": "phase=interval value99;n=99",
      "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": "S072",
      "uses": [
        "G001",
        "S071"
      ],
      "before": "interval value99 trial divisor2:natural dividend=99;positive divisor=2",
      "after": "99=49·2+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": [
          "99",
          "2",
          "49"
        ],
        "output": "1",
        "label": "interval value99 trial divisor2"
      }
    },
    {
      "id": "S073",
      "uses": [
        "G001",
        "S072"
      ],
      "before": "interval value99 trial divisor3:natural dividend=99;positive divisor=3",
      "after": "99=33·3+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": [
          "99",
          "3",
          "33"
        ],
        "output": "0",
        "label": "interval value99 trial divisor3"
      }
    },
    {
      "id": "S074",
      "uses": [
        "G001",
        "S073"
      ],
      "before": "phase=interval value99;tried ascending divisors=[2, 3];sqrt floor=9",
      "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": "S075",
      "uses": [
        "G001",
        "S074"
      ],
      "before": "phase=interval value100;n=100",
      "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": "S076",
      "uses": [
        "G001",
        "S075"
      ],
      "before": "interval value100 trial divisor2:natural dividend=100;positive divisor=2",
      "after": "100=50·2+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": [
          "100",
          "2",
          "50"
        ],
        "output": "0",
        "label": "interval value100 trial divisor2"
      }
    },
    {
      "id": "S077",
      "uses": [
        "G001",
        "S076"
      ],
      "before": "phase=interval value100;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": "S078",
      "uses": [
        "G001",
        "S077"
      ],
      "before": "phase=interval value101;n=101",
      "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": "S079",
      "uses": [
        "G001",
        "S078"
      ],
      "before": "interval value101 trial divisor2:natural dividend=101;positive divisor=2",
      "after": "101=50·2+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": [
          "101",
          "2",
          "50"
        ],
        "output": "1",
        "label": "interval value101 trial divisor2"
      }
    },
    {
      "id": "S080",
      "uses": [
        "G001",
        "S079"
      ],
      "before": "interval value101 trial divisor3:natural dividend=101;positive divisor=3",
      "after": "101=33·3+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": [
          "101",
          "3",
          "33"
        ],
        "output": "2",
        "label": "interval value101 trial divisor3"
      }
    },
    {
      "id": "S081",
      "uses": [
        "G001",
        "S080"
      ],
      "before": "interval value101 trial divisor4:natural dividend=101;positive divisor=4",
      "after": "101=25·4+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": [
          "101",
          "4",
          "25"
        ],
        "output": "1",
        "label": "interval value101 trial divisor4"
      }
    },
    {
      "id": "S082",
      "uses": [
        "G001",
        "S081"
      ],
      "before": "interval value101 trial divisor5:natural dividend=101;positive divisor=5",
      "after": "101=20·5+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": [
          "101",
          "5",
          "20"
        ],
        "output": "1",
        "label": "interval value101 trial divisor5"
      }
    },
    {
      "id": "S083",
      "uses": [
        "G001",
        "S082"
      ],
      "before": "interval value101 trial divisor6:natural dividend=101;positive divisor=6",
      "after": "101=16·6+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": [
          "101",
          "6",
          "16"
        ],
        "output": "5",
        "label": "interval value101 trial divisor6"
      }
    },
    {
      "id": "S084",
      "uses": [
        "G001",
        "S083"
      ],
      "before": "interval value101 trial divisor7:natural dividend=101;positive divisor=7",
      "after": "101=14·7+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": [
          "101",
          "7",
          "14"
        ],
        "output": "3",
        "label": "interval value101 trial divisor7"
      }
    },
    {
      "id": "S085",
      "uses": [
        "G001",
        "S084"
      ],
      "before": "interval value101 trial divisor8:natural dividend=101;positive divisor=8",
      "after": "101=12·8+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": [
          "101",
          "8",
          "12"
        ],
        "output": "5",
        "label": "interval value101 trial divisor8"
      }
    },
    {
      "id": "S086",
      "uses": [
        "G001",
        "S085"
      ],
      "before": "interval value101 trial divisor9:natural dividend=101;positive divisor=9",
      "after": "101=11·9+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": [
          "101",
          "9",
          "11"
        ],
        "output": "2",
        "label": "interval value101 trial divisor9"
      }
    },
    {
      "id": "S087",
      "uses": [
        "G001",
        "S086"
      ],
      "before": "interval value101 trial divisor10:natural dividend=101;positive divisor=10",
      "after": "101=10·10+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": [
          "101",
          "10",
          "10"
        ],
        "output": "1",
        "label": "interval value101 trial divisor10"
      }
    },
    {
      "id": "S088",
      "uses": [
        "G001",
        "S087"
      ],
      "before": "phase=interval value101;tried ascending divisors=[2, 3, 4, 5, 6, 7, 8, 9, 10];sqrt floor=10",
      "after": "least factor at least2=101;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": "S089",
      "uses": [
        "G001",
        "S088"
      ],
      "before": "phase=interval value102;n=102",
      "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": "S090",
      "uses": [
        "G001",
        "S089"
      ],
      "before": "interval value102 trial divisor2:natural dividend=102;positive divisor=2",
      "after": "102=51·2+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": [
          "102",
          "2",
          "51"
        ],
        "output": "0",
        "label": "interval value102 trial divisor2"
      }
    },
    {
      "id": "S091",
      "uses": [
        "G001",
        "S090"
      ],
      "before": "phase=interval value102;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": "S092",
      "uses": [
        "G001",
        "S091"
      ],
      "before": "phase=interval value103;n=103",
      "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": "S093",
      "uses": [
        "G001",
        "S092"
      ],
      "before": "interval value103 trial divisor2:natural dividend=103;positive divisor=2",
      "after": "103=51·2+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": [
          "103",
          "2",
          "51"
        ],
        "output": "1",
        "label": "interval value103 trial divisor2"
      }
    },
    {
      "id": "S094",
      "uses": [
        "G001",
        "S093"
      ],
      "before": "interval value103 trial divisor3:natural dividend=103;positive divisor=3",
      "after": "103=34·3+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": [
          "103",
          "3",
          "34"
        ],
        "output": "1",
        "label": "interval value103 trial divisor3"
      }
    },
    {
      "id": "S095",
      "uses": [
        "G001",
        "S094"
      ],
      "before": "interval value103 trial divisor4:natural dividend=103;positive divisor=4",
      "after": "103=25·4+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": [
          "103",
          "4",
          "25"
        ],
        "output": "3",
        "label": "interval value103 trial divisor4"
      }
    },
    {
      "id": "S096",
      "uses": [
        "G001",
        "S095"
      ],
      "before": "interval value103 trial divisor5:natural dividend=103;positive divisor=5",
      "after": "103=20·5+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": [
          "103",
          "5",
          "20"
        ],
        "output": "3",
        "label": "interval value103 trial divisor5"
      }
    },
    {
      "id": "S097",
      "uses": [
        "G001",
        "S096"
      ],
      "before": "interval value103 trial divisor6:natural dividend=103;positive divisor=6",
      "after": "103=17·6+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": [
          "103",
          "6",
          "17"
        ],
        "output": "1",
        "label": "interval value103 trial divisor6"
      }
    },
    {
      "id": "S098",
      "uses": [
        "G001",
        "S097"
      ],
      "before": "interval value103 trial divisor7:natural dividend=103;positive divisor=7",
      "after": "103=14·7+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": [
          "103",
          "7",
          "14"
        ],
        "output": "5",
        "label": "interval value103 trial divisor7"
      }
    },
    {
      "id": "S099",
      "uses": [
        "G001",
        "S098"
      ],
      "before": "interval value103 trial divisor8:natural dividend=103;positive divisor=8",
      "after": "103=12·8+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": [
          "103",
          "8",
          "12"
        ],
        "output": "7",
        "label": "interval value103 trial divisor8"
      }
    },
    {
      "id": "S100",
      "uses": [
        "G001",
        "S099"
      ],
      "before": "interval value103 trial divisor9:natural dividend=103;positive divisor=9",
      "after": "103=11·9+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": [
          "103",
          "9",
          "11"
        ],
        "output": "4",
        "label": "interval value103 trial divisor9"
      }
    },
    {
      "id": "S101",
      "uses": [
        "G001",
        "S100"
      ],
      "before": "interval value103 trial divisor10:natural dividend=103;positive divisor=10",
      "after": "103=10·10+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": [
          "103",
          "10",
          "10"
        ],
        "output": "3",
        "label": "interval value103 trial divisor10"
      }
    },
    {
      "id": "S102",
      "uses": [
        "G001",
        "S101"
      ],
      "before": "phase=interval value103;tried ascending divisors=[2, 3, 4, 5, 6, 7, 8, 9, 10];sqrt floor=10",
      "after": "least factor at least2=103;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 value104;n=104",
      "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 value104 trial divisor2:natural dividend=104;positive divisor=2",
      "after": "104=52·2+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": [
          "104",
          "2",
          "52"
        ],
        "output": "0",
        "label": "interval value104 trial divisor2"
      }
    },
    {
      "id": "S105",
      "uses": [
        "G001",
        "S104"
      ],
      "before": "phase=interval value104;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": "S106",
      "uses": [
        "G001",
        "S105"
      ],
      "before": "phase=interval value105;n=105",
      "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 value105 trial divisor2:natural dividend=105;positive divisor=2",
      "after": "105=52·2+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": [
          "105",
          "2",
          "52"
        ],
        "output": "1",
        "label": "interval value105 trial divisor2"
      }
    },
    {
      "id": "S108",
      "uses": [
        "G001",
        "S107"
      ],
      "before": "interval value105 trial divisor3:natural dividend=105;positive divisor=3",
      "after": "105=35·3+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": [
          "105",
          "3",
          "35"
        ],
        "output": "0",
        "label": "interval value105 trial divisor3"
      }
    },
    {
      "id": "S109",
      "uses": [
        "G001",
        "S108"
      ],
      "before": "phase=interval value105;tried ascending divisors=[2, 3];sqrt floor=10",
      "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": "S110",
      "uses": [
        "G001",
        "S109"
      ],
      "before": "inclusive interval=[85, 105]",
      "after": "all primes=[89, 97, 101, 103];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": [
      85,
      105,
      [
        89,
        97,
        101,
        103
      ],
      4
    ]
  },
  "conclusion": {
    "canonical_value": "(85, 105, [89, 97, 101, 103], 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": 93,
      "original_line_sha256": "d5f1ce216257d991ac404f1f2cfff2f8e3a7a79ad6f428c3fc42cd9ebe3c1c6b",
      "question_compact_sha256": "2318c0290f9c140adec69313a93596042ddebdfaa6fc213cc745667064cd7708",
      "english_row_compact_sha256": "00fbd8f7d4459abffdd737cb424a4bdf07913d98b46f8e486d485f890ab26c0d",
      "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": "2b94962781630e1df7f67cd21ecde761520c6eae454aee6b613ba2b0e7f08276",
    "original_candidate_line_sha256": "2b94962781630e1df7f67cd21ecde761520c6eae454aee6b613ba2b0e7f08276"
  },
  "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": 110,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "4ec3bf306e03eed05f9b1ee9fea530e22fb503ef8369ff0a6509ba0755ce4476",
      "arithmetic_steps": 66
    },
    "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."
  }
}