{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261013-advance-20261018-r04-prime-number-0088",
  "pilot_number": "R0017-09163",
  "number": 9163,
  "family_id": "qf-ext18-prime-interval-inventory",
  "title": {
    "zh-Hant": "第 9163 題：自然數質數與因數",
    "en": "Question 9163: Every Prime in an Inclusive Bounded Interval"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261013-advance-20261018-r04-prime-number-0088",
    "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\":60,\"hi\":80}。請依 [lo,hi,primes,count] 的欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(60, 80, [61, 67, 71, 73, 79], 5)",
      "explanation_zh": "擴展基礎核對=(60, 80, [61, 67, 71, 73, 79], 5)；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": 2,
      "skills": [
        "自然數質數與因數"
      ]
    },
    "variant": {
      "family_id": "qf-ext18-prime-interval-inventory",
      "seed": 202610184088,
      "parameters": {
        "lo": 60,
        "hi": 80,
        "domain": "Nat_prime",
        "operation": "prime-interval-inventory",
        "semantic_template": "qf-ext18-prime-interval-inventory/{\"hi\":80,\"lo\":60}"
      }
    },
    "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-0088",
    "batch_id": "batch-20261013-advance-20261018-r04-prime-number-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "e7f423eaea0b406bba8f8efda7a9088024223370364c2538bbea3da35c5c573a",
      "family_id": "qf-ext18-prime-interval-inventory",
      "seed": 202610184088,
      "semantic_template": "qf-ext18-prime-interval-inventory/{\"hi\":80,\"lo\":60}"
    },
    "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=60, hi=80, with lo≤hi and a bounded window. Give (lo, hi, every prime in [lo,hi] in increasing order, count).",
      "answer": {
        "kind": "expression",
        "value": "(60, 80, [61, 67, 71, 73, 79], 5)",
        "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 (60, 80, [61, 67, 71, 73, 79], 5)."
      },
      "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\":60,\"hi\":80}。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=60, hi=80, 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 value60;n=60",
      "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 value60 trial divisor2:natural dividend=60;positive divisor=2",
      "after": "60=30·2+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": [
          "60",
          "2",
          "30"
        ],
        "output": "0",
        "label": "interval value60 trial divisor2"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "phase=interval value60;tried ascending divisors=[2];sqrt floor=7",
      "after": "least factor at least2=2;prime=否",
      "relation": "implies",
      "rule_id": "complete_small_factor_bound",
      "conditions": [],
      "explanation": {
        "zh-Hant": "若n=uv合數且u,v≥2，至少一因子≤sqrt(n)。依序找到首個零餘數就是最小因子；完整界內皆無因子時n自己是最小至少2因子，故為質數。",
        "en": "If n=uv is composite with u,v≥2,one factor is≤sqrt(n).The first zero remainder in ascending order gives the least factor;if none exists through the full bound,n itself is the least factor≥2 and is prime."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "phase=interval value61;n=61",
      "after": "n>=2=是",
      "relation": "implies",
      "rule_id": "prime_lower_bound_guard",
      "conditions": [],
      "explanation": {
        "zh-Hant": "質數必須至少2；0與1先排除，不能把沒有非平凡試除因子誤當質數。",
        "en": "A prime must be at least2.Exclude0 and1 first rather than treating an absence of trial factors as primality."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "interval value61 trial divisor2:natural dividend=61;positive divisor=2",
      "after": "61=30·2+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": [
          "61",
          "2",
          "30"
        ],
        "output": "1",
        "label": "interval value61 trial divisor2"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "interval value61 trial divisor3:natural dividend=61;positive divisor=3",
      "after": "61=20·3+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": [
          "61",
          "3",
          "20"
        ],
        "output": "1",
        "label": "interval value61 trial divisor3"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "interval value61 trial divisor4:natural dividend=61;positive divisor=4",
      "after": "61=15·4+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": [
          "61",
          "4",
          "15"
        ],
        "output": "1",
        "label": "interval value61 trial divisor4"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "interval value61 trial divisor5:natural dividend=61;positive divisor=5",
      "after": "61=12·5+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": [
          "61",
          "5",
          "12"
        ],
        "output": "1",
        "label": "interval value61 trial divisor5"
      }
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "interval value61 trial divisor6:natural dividend=61;positive divisor=6",
      "after": "61=10·6+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": [
          "61",
          "6",
          "10"
        ],
        "output": "1",
        "label": "interval value61 trial divisor6"
      }
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "interval value61 trial divisor7:natural dividend=61;positive divisor=7",
      "after": "61=8·7+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": [
          "61",
          "7",
          "8"
        ],
        "output": "5",
        "label": "interval value61 trial divisor7"
      }
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "phase=interval value61;tried ascending divisors=[2, 3, 4, 5, 6, 7];sqrt floor=7",
      "after": "least factor at least2=61;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": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "phase=interval value62;n=62",
      "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": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "interval value62 trial divisor2:natural dividend=62;positive divisor=2",
      "after": "62=31·2+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": [
          "62",
          "2",
          "31"
        ],
        "output": "0",
        "label": "interval value62 trial divisor2"
      }
    },
    {
      "id": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "phase=interval value62;tried ascending divisors=[2];sqrt floor=7",
      "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": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "phase=interval value63;n=63",
      "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": "S017",
      "uses": [
        "G001",
        "S016"
      ],
      "before": "interval value63 trial divisor2:natural dividend=63;positive divisor=2",
      "after": "63=31·2+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": [
          "63",
          "2",
          "31"
        ],
        "output": "1",
        "label": "interval value63 trial divisor2"
      }
    },
    {
      "id": "S018",
      "uses": [
        "G001",
        "S017"
      ],
      "before": "interval value63 trial divisor3:natural dividend=63;positive divisor=3",
      "after": "63=21·3+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": [
          "63",
          "3",
          "21"
        ],
        "output": "0",
        "label": "interval value63 trial divisor3"
      }
    },
    {
      "id": "S019",
      "uses": [
        "G001",
        "S018"
      ],
      "before": "phase=interval value63;tried ascending divisors=[2, 3];sqrt floor=7",
      "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": "S020",
      "uses": [
        "G001",
        "S019"
      ],
      "before": "phase=interval value64;n=64",
      "after": "n>=2=是",
      "relation": "implies",
      "rule_id": "prime_lower_bound_guard",
      "conditions": [],
      "explanation": {
        "zh-Hant": "質數必須至少2；0與1先排除，不能把沒有非平凡試除因子誤當質數。",
        "en": "A prime must be at least2.Exclude0 and1 first rather than treating an absence of trial factors as primality."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S021",
      "uses": [
        "G001",
        "S020"
      ],
      "before": "interval value64 trial divisor2:natural dividend=64;positive divisor=2",
      "after": "64=32·2+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": [
          "64",
          "2",
          "32"
        ],
        "output": "0",
        "label": "interval value64 trial divisor2"
      }
    },
    {
      "id": "S022",
      "uses": [
        "G001",
        "S021"
      ],
      "before": "phase=interval value64;tried ascending divisors=[2];sqrt floor=8",
      "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": "S023",
      "uses": [
        "G001",
        "S022"
      ],
      "before": "phase=interval value65;n=65",
      "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": "S024",
      "uses": [
        "G001",
        "S023"
      ],
      "before": "interval value65 trial divisor2:natural dividend=65;positive divisor=2",
      "after": "65=32·2+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": [
          "65",
          "2",
          "32"
        ],
        "output": "1",
        "label": "interval value65 trial divisor2"
      }
    },
    {
      "id": "S025",
      "uses": [
        "G001",
        "S024"
      ],
      "before": "interval value65 trial divisor3:natural dividend=65;positive divisor=3",
      "after": "65=21·3+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": [
          "65",
          "3",
          "21"
        ],
        "output": "2",
        "label": "interval value65 trial divisor3"
      }
    },
    {
      "id": "S026",
      "uses": [
        "G001",
        "S025"
      ],
      "before": "interval value65 trial divisor4:natural dividend=65;positive divisor=4",
      "after": "65=16·4+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": [
          "65",
          "4",
          "16"
        ],
        "output": "1",
        "label": "interval value65 trial divisor4"
      }
    },
    {
      "id": "S027",
      "uses": [
        "G001",
        "S026"
      ],
      "before": "interval value65 trial divisor5:natural dividend=65;positive divisor=5",
      "after": "65=13·5+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": [
          "65",
          "5",
          "13"
        ],
        "output": "0",
        "label": "interval value65 trial divisor5"
      }
    },
    {
      "id": "S028",
      "uses": [
        "G001",
        "S027"
      ],
      "before": "phase=interval value65;tried ascending divisors=[2, 3, 4, 5];sqrt floor=8",
      "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": "S029",
      "uses": [
        "G001",
        "S028"
      ],
      "before": "phase=interval value66;n=66",
      "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": "S030",
      "uses": [
        "G001",
        "S029"
      ],
      "before": "interval value66 trial divisor2:natural dividend=66;positive divisor=2",
      "after": "66=33·2+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": [
          "66",
          "2",
          "33"
        ],
        "output": "0",
        "label": "interval value66 trial divisor2"
      }
    },
    {
      "id": "S031",
      "uses": [
        "G001",
        "S030"
      ],
      "before": "phase=interval value66;tried ascending divisors=[2];sqrt floor=8",
      "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": "S032",
      "uses": [
        "G001",
        "S031"
      ],
      "before": "phase=interval value67;n=67",
      "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": "S033",
      "uses": [
        "G001",
        "S032"
      ],
      "before": "interval value67 trial divisor2:natural dividend=67;positive divisor=2",
      "after": "67=33·2+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": [
          "67",
          "2",
          "33"
        ],
        "output": "1",
        "label": "interval value67 trial divisor2"
      }
    },
    {
      "id": "S034",
      "uses": [
        "G001",
        "S033"
      ],
      "before": "interval value67 trial divisor3:natural dividend=67;positive divisor=3",
      "after": "67=22·3+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": [
          "67",
          "3",
          "22"
        ],
        "output": "1",
        "label": "interval value67 trial divisor3"
      }
    },
    {
      "id": "S035",
      "uses": [
        "G001",
        "S034"
      ],
      "before": "interval value67 trial divisor4:natural dividend=67;positive divisor=4",
      "after": "67=16·4+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": [
          "67",
          "4",
          "16"
        ],
        "output": "3",
        "label": "interval value67 trial divisor4"
      }
    },
    {
      "id": "S036",
      "uses": [
        "G001",
        "S035"
      ],
      "before": "interval value67 trial divisor5:natural dividend=67;positive divisor=5",
      "after": "67=13·5+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": [
          "67",
          "5",
          "13"
        ],
        "output": "2",
        "label": "interval value67 trial divisor5"
      }
    },
    {
      "id": "S037",
      "uses": [
        "G001",
        "S036"
      ],
      "before": "interval value67 trial divisor6:natural dividend=67;positive divisor=6",
      "after": "67=11·6+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": [
          "67",
          "6",
          "11"
        ],
        "output": "1",
        "label": "interval value67 trial divisor6"
      }
    },
    {
      "id": "S038",
      "uses": [
        "G001",
        "S037"
      ],
      "before": "interval value67 trial divisor7:natural dividend=67;positive divisor=7",
      "after": "67=9·7+4",
      "relation": "implies",
      "rule_id": "positive_euclidean_division",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數下完整列商餘等式，0≤r<divisor；不把取模0當可求值。",
        "en": "With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": {
        "operation": "euclidean_division",
        "inputs": [
          "67",
          "7",
          "9"
        ],
        "output": "4",
        "label": "interval value67 trial divisor7"
      }
    },
    {
      "id": "S039",
      "uses": [
        "G001",
        "S038"
      ],
      "before": "interval value67 trial divisor8:natural dividend=67;positive divisor=8",
      "after": "67=8·8+3",
      "relation": "implies",
      "rule_id": "positive_euclidean_division",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數下完整列商餘等式，0≤r<divisor；不把取模0當可求值。",
        "en": "With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": {
        "operation": "euclidean_division",
        "inputs": [
          "67",
          "8",
          "8"
        ],
        "output": "3",
        "label": "interval value67 trial divisor8"
      }
    },
    {
      "id": "S040",
      "uses": [
        "G001",
        "S039"
      ],
      "before": "phase=interval value67;tried ascending divisors=[2, 3, 4, 5, 6, 7, 8];sqrt floor=8",
      "after": "least factor at least2=67;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": "S041",
      "uses": [
        "G001",
        "S040"
      ],
      "before": "phase=interval value68;n=68",
      "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": "S042",
      "uses": [
        "G001",
        "S041"
      ],
      "before": "interval value68 trial divisor2:natural dividend=68;positive divisor=2",
      "after": "68=34·2+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": [
          "68",
          "2",
          "34"
        ],
        "output": "0",
        "label": "interval value68 trial divisor2"
      }
    },
    {
      "id": "S043",
      "uses": [
        "G001",
        "S042"
      ],
      "before": "phase=interval value68;tried ascending divisors=[2];sqrt floor=8",
      "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": "S044",
      "uses": [
        "G001",
        "S043"
      ],
      "before": "phase=interval value69;n=69",
      "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": "S045",
      "uses": [
        "G001",
        "S044"
      ],
      "before": "interval value69 trial divisor2:natural dividend=69;positive divisor=2",
      "after": "69=34·2+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": [
          "69",
          "2",
          "34"
        ],
        "output": "1",
        "label": "interval value69 trial divisor2"
      }
    },
    {
      "id": "S046",
      "uses": [
        "G001",
        "S045"
      ],
      "before": "interval value69 trial divisor3:natural dividend=69;positive divisor=3",
      "after": "69=23·3+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": [
          "69",
          "3",
          "23"
        ],
        "output": "0",
        "label": "interval value69 trial divisor3"
      }
    },
    {
      "id": "S047",
      "uses": [
        "G001",
        "S046"
      ],
      "before": "phase=interval value69;tried ascending divisors=[2, 3];sqrt floor=8",
      "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": "S048",
      "uses": [
        "G001",
        "S047"
      ],
      "before": "phase=interval value70;n=70",
      "after": "n>=2=是",
      "relation": "implies",
      "rule_id": "prime_lower_bound_guard",
      "conditions": [],
      "explanation": {
        "zh-Hant": "質數必須至少2；0與1先排除，不能把沒有非平凡試除因子誤當質數。",
        "en": "A prime must be at least2.Exclude0 and1 first rather than treating an absence of trial factors as primality."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S049",
      "uses": [
        "G001",
        "S048"
      ],
      "before": "interval value70 trial divisor2:natural dividend=70;positive divisor=2",
      "after": "70=35·2+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": [
          "70",
          "2",
          "35"
        ],
        "output": "0",
        "label": "interval value70 trial divisor2"
      }
    },
    {
      "id": "S050",
      "uses": [
        "G001",
        "S049"
      ],
      "before": "phase=interval value70;tried ascending divisors=[2];sqrt floor=8",
      "after": "least factor at least2=2;prime=否",
      "relation": "implies",
      "rule_id": "complete_small_factor_bound",
      "conditions": [],
      "explanation": {
        "zh-Hant": "若n=uv合數且u,v≥2，至少一因子≤sqrt(n)。依序找到首個零餘數就是最小因子；完整界內皆無因子時n自己是最小至少2因子，故為質數。",
        "en": "If n=uv is composite with u,v≥2,one factor is≤sqrt(n).The first zero remainder in ascending order gives the least factor;if none exists through the full bound,n itself is the least factor≥2 and is prime."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S051",
      "uses": [
        "G001",
        "S050"
      ],
      "before": "phase=interval value71;n=71",
      "after": "n>=2=是",
      "relation": "implies",
      "rule_id": "prime_lower_bound_guard",
      "conditions": [],
      "explanation": {
        "zh-Hant": "質數必須至少2；0與1先排除，不能把沒有非平凡試除因子誤當質數。",
        "en": "A prime must be at least2.Exclude0 and1 first rather than treating an absence of trial factors as primality."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S052",
      "uses": [
        "G001",
        "S051"
      ],
      "before": "interval value71 trial divisor2:natural dividend=71;positive divisor=2",
      "after": "71=35·2+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": [
          "71",
          "2",
          "35"
        ],
        "output": "1",
        "label": "interval value71 trial divisor2"
      }
    },
    {
      "id": "S053",
      "uses": [
        "G001",
        "S052"
      ],
      "before": "interval value71 trial divisor3:natural dividend=71;positive divisor=3",
      "after": "71=23·3+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": [
          "71",
          "3",
          "23"
        ],
        "output": "2",
        "label": "interval value71 trial divisor3"
      }
    },
    {
      "id": "S054",
      "uses": [
        "G001",
        "S053"
      ],
      "before": "interval value71 trial divisor4:natural dividend=71;positive divisor=4",
      "after": "71=17·4+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": [
          "71",
          "4",
          "17"
        ],
        "output": "3",
        "label": "interval value71 trial divisor4"
      }
    },
    {
      "id": "S055",
      "uses": [
        "G001",
        "S054"
      ],
      "before": "interval value71 trial divisor5:natural dividend=71;positive divisor=5",
      "after": "71=14·5+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": [
          "71",
          "5",
          "14"
        ],
        "output": "1",
        "label": "interval value71 trial divisor5"
      }
    },
    {
      "id": "S056",
      "uses": [
        "G001",
        "S055"
      ],
      "before": "interval value71 trial divisor6:natural dividend=71;positive divisor=6",
      "after": "71=11·6+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": [
          "71",
          "6",
          "11"
        ],
        "output": "5",
        "label": "interval value71 trial divisor6"
      }
    },
    {
      "id": "S057",
      "uses": [
        "G001",
        "S056"
      ],
      "before": "interval value71 trial divisor7:natural dividend=71;positive divisor=7",
      "after": "71=10·7+1",
      "relation": "implies",
      "rule_id": "positive_euclidean_division",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數下完整列商餘等式，0≤r<divisor；不把取模0當可求值。",
        "en": "With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": {
        "operation": "euclidean_division",
        "inputs": [
          "71",
          "7",
          "10"
        ],
        "output": "1",
        "label": "interval value71 trial divisor7"
      }
    },
    {
      "id": "S058",
      "uses": [
        "G001",
        "S057"
      ],
      "before": "interval value71 trial divisor8:natural dividend=71;positive divisor=8",
      "after": "71=8·8+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": [
          "71",
          "8",
          "8"
        ],
        "output": "7",
        "label": "interval value71 trial divisor8"
      }
    },
    {
      "id": "S059",
      "uses": [
        "G001",
        "S058"
      ],
      "before": "phase=interval value71;tried ascending divisors=[2, 3, 4, 5, 6, 7, 8];sqrt floor=8",
      "after": "least factor at least2=71;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": "S060",
      "uses": [
        "G001",
        "S059"
      ],
      "before": "phase=interval value72;n=72",
      "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": "S061",
      "uses": [
        "G001",
        "S060"
      ],
      "before": "interval value72 trial divisor2:natural dividend=72;positive divisor=2",
      "after": "72=36·2+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": [
          "72",
          "2",
          "36"
        ],
        "output": "0",
        "label": "interval value72 trial divisor2"
      }
    },
    {
      "id": "S062",
      "uses": [
        "G001",
        "S061"
      ],
      "before": "phase=interval value72;tried ascending divisors=[2];sqrt floor=8",
      "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": "S063",
      "uses": [
        "G001",
        "S062"
      ],
      "before": "phase=interval value73;n=73",
      "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": "S064",
      "uses": [
        "G001",
        "S063"
      ],
      "before": "interval value73 trial divisor2:natural dividend=73;positive divisor=2",
      "after": "73=36·2+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": [
          "73",
          "2",
          "36"
        ],
        "output": "1",
        "label": "interval value73 trial divisor2"
      }
    },
    {
      "id": "S065",
      "uses": [
        "G001",
        "S064"
      ],
      "before": "interval value73 trial divisor3:natural dividend=73;positive divisor=3",
      "after": "73=24·3+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": [
          "73",
          "3",
          "24"
        ],
        "output": "1",
        "label": "interval value73 trial divisor3"
      }
    },
    {
      "id": "S066",
      "uses": [
        "G001",
        "S065"
      ],
      "before": "interval value73 trial divisor4:natural dividend=73;positive divisor=4",
      "after": "73=18·4+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": [
          "73",
          "4",
          "18"
        ],
        "output": "1",
        "label": "interval value73 trial divisor4"
      }
    },
    {
      "id": "S067",
      "uses": [
        "G001",
        "S066"
      ],
      "before": "interval value73 trial divisor5:natural dividend=73;positive divisor=5",
      "after": "73=14·5+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": [
          "73",
          "5",
          "14"
        ],
        "output": "3",
        "label": "interval value73 trial divisor5"
      }
    },
    {
      "id": "S068",
      "uses": [
        "G001",
        "S067"
      ],
      "before": "interval value73 trial divisor6:natural dividend=73;positive divisor=6",
      "after": "73=12·6+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": [
          "73",
          "6",
          "12"
        ],
        "output": "1",
        "label": "interval value73 trial divisor6"
      }
    },
    {
      "id": "S069",
      "uses": [
        "G001",
        "S068"
      ],
      "before": "interval value73 trial divisor7:natural dividend=73;positive divisor=7",
      "after": "73=10·7+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": [
          "73",
          "7",
          "10"
        ],
        "output": "3",
        "label": "interval value73 trial divisor7"
      }
    },
    {
      "id": "S070",
      "uses": [
        "G001",
        "S069"
      ],
      "before": "interval value73 trial divisor8:natural dividend=73;positive divisor=8",
      "after": "73=9·8+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": [
          "73",
          "8",
          "9"
        ],
        "output": "1",
        "label": "interval value73 trial divisor8"
      }
    },
    {
      "id": "S071",
      "uses": [
        "G001",
        "S070"
      ],
      "before": "phase=interval value73;tried ascending divisors=[2, 3, 4, 5, 6, 7, 8];sqrt floor=8",
      "after": "least factor at least2=73;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": "S072",
      "uses": [
        "G001",
        "S071"
      ],
      "before": "phase=interval value74;n=74",
      "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": "S073",
      "uses": [
        "G001",
        "S072"
      ],
      "before": "interval value74 trial divisor2:natural dividend=74;positive divisor=2",
      "after": "74=37·2+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": [
          "74",
          "2",
          "37"
        ],
        "output": "0",
        "label": "interval value74 trial divisor2"
      }
    },
    {
      "id": "S074",
      "uses": [
        "G001",
        "S073"
      ],
      "before": "phase=interval value74;tried ascending divisors=[2];sqrt floor=8",
      "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": "S075",
      "uses": [
        "G001",
        "S074"
      ],
      "before": "phase=interval value75;n=75",
      "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 value75 trial divisor2:natural dividend=75;positive divisor=2",
      "after": "75=37·2+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": [
          "75",
          "2",
          "37"
        ],
        "output": "1",
        "label": "interval value75 trial divisor2"
      }
    },
    {
      "id": "S077",
      "uses": [
        "G001",
        "S076"
      ],
      "before": "interval value75 trial divisor3:natural dividend=75;positive divisor=3",
      "after": "75=25·3+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": [
          "75",
          "3",
          "25"
        ],
        "output": "0",
        "label": "interval value75 trial divisor3"
      }
    },
    {
      "id": "S078",
      "uses": [
        "G001",
        "S077"
      ],
      "before": "phase=interval value75;tried ascending divisors=[2, 3];sqrt floor=8",
      "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": "S079",
      "uses": [
        "G001",
        "S078"
      ],
      "before": "phase=interval value76;n=76",
      "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": "S080",
      "uses": [
        "G001",
        "S079"
      ],
      "before": "interval value76 trial divisor2:natural dividend=76;positive divisor=2",
      "after": "76=38·2+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": [
          "76",
          "2",
          "38"
        ],
        "output": "0",
        "label": "interval value76 trial divisor2"
      }
    },
    {
      "id": "S081",
      "uses": [
        "G001",
        "S080"
      ],
      "before": "phase=interval value76;tried ascending divisors=[2];sqrt floor=8",
      "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": "S082",
      "uses": [
        "G001",
        "S081"
      ],
      "before": "phase=interval value77;n=77",
      "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": "S083",
      "uses": [
        "G001",
        "S082"
      ],
      "before": "interval value77 trial divisor2:natural dividend=77;positive divisor=2",
      "after": "77=38·2+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": [
          "77",
          "2",
          "38"
        ],
        "output": "1",
        "label": "interval value77 trial divisor2"
      }
    },
    {
      "id": "S084",
      "uses": [
        "G001",
        "S083"
      ],
      "before": "interval value77 trial divisor3:natural dividend=77;positive divisor=3",
      "after": "77=25·3+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": [
          "77",
          "3",
          "25"
        ],
        "output": "2",
        "label": "interval value77 trial divisor3"
      }
    },
    {
      "id": "S085",
      "uses": [
        "G001",
        "S084"
      ],
      "before": "interval value77 trial divisor4:natural dividend=77;positive divisor=4",
      "after": "77=19·4+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": [
          "77",
          "4",
          "19"
        ],
        "output": "1",
        "label": "interval value77 trial divisor4"
      }
    },
    {
      "id": "S086",
      "uses": [
        "G001",
        "S085"
      ],
      "before": "interval value77 trial divisor5:natural dividend=77;positive divisor=5",
      "after": "77=15·5+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": [
          "77",
          "5",
          "15"
        ],
        "output": "2",
        "label": "interval value77 trial divisor5"
      }
    },
    {
      "id": "S087",
      "uses": [
        "G001",
        "S086"
      ],
      "before": "interval value77 trial divisor6:natural dividend=77;positive divisor=6",
      "after": "77=12·6+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": [
          "77",
          "6",
          "12"
        ],
        "output": "5",
        "label": "interval value77 trial divisor6"
      }
    },
    {
      "id": "S088",
      "uses": [
        "G001",
        "S087"
      ],
      "before": "interval value77 trial divisor7:natural dividend=77;positive divisor=7",
      "after": "77=11·7+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": [
          "77",
          "7",
          "11"
        ],
        "output": "0",
        "label": "interval value77 trial divisor7"
      }
    },
    {
      "id": "S089",
      "uses": [
        "G001",
        "S088"
      ],
      "before": "phase=interval value77;tried ascending divisors=[2, 3, 4, 5, 6, 7];sqrt floor=8",
      "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": "S090",
      "uses": [
        "G001",
        "S089"
      ],
      "before": "phase=interval value78;n=78",
      "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": "S091",
      "uses": [
        "G001",
        "S090"
      ],
      "before": "interval value78 trial divisor2:natural dividend=78;positive divisor=2",
      "after": "78=39·2+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": [
          "78",
          "2",
          "39"
        ],
        "output": "0",
        "label": "interval value78 trial divisor2"
      }
    },
    {
      "id": "S092",
      "uses": [
        "G001",
        "S091"
      ],
      "before": "phase=interval value78;tried ascending divisors=[2];sqrt floor=8",
      "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": "S093",
      "uses": [
        "G001",
        "S092"
      ],
      "before": "phase=interval value79;n=79",
      "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": "S094",
      "uses": [
        "G001",
        "S093"
      ],
      "before": "interval value79 trial divisor2:natural dividend=79;positive divisor=2",
      "after": "79=39·2+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": [
          "79",
          "2",
          "39"
        ],
        "output": "1",
        "label": "interval value79 trial divisor2"
      }
    },
    {
      "id": "S095",
      "uses": [
        "G001",
        "S094"
      ],
      "before": "interval value79 trial divisor3:natural dividend=79;positive divisor=3",
      "after": "79=26·3+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": [
          "79",
          "3",
          "26"
        ],
        "output": "1",
        "label": "interval value79 trial divisor3"
      }
    },
    {
      "id": "S096",
      "uses": [
        "G001",
        "S095"
      ],
      "before": "interval value79 trial divisor4:natural dividend=79;positive divisor=4",
      "after": "79=19·4+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": [
          "79",
          "4",
          "19"
        ],
        "output": "3",
        "label": "interval value79 trial divisor4"
      }
    },
    {
      "id": "S097",
      "uses": [
        "G001",
        "S096"
      ],
      "before": "interval value79 trial divisor5:natural dividend=79;positive divisor=5",
      "after": "79=15·5+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": [
          "79",
          "5",
          "15"
        ],
        "output": "4",
        "label": "interval value79 trial divisor5"
      }
    },
    {
      "id": "S098",
      "uses": [
        "G001",
        "S097"
      ],
      "before": "interval value79 trial divisor6:natural dividend=79;positive divisor=6",
      "after": "79=13·6+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": [
          "79",
          "6",
          "13"
        ],
        "output": "1",
        "label": "interval value79 trial divisor6"
      }
    },
    {
      "id": "S099",
      "uses": [
        "G001",
        "S098"
      ],
      "before": "interval value79 trial divisor7:natural dividend=79;positive divisor=7",
      "after": "79=11·7+2",
      "relation": "implies",
      "rule_id": "positive_euclidean_division",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數下完整列商餘等式，0≤r<divisor；不把取模0當可求值。",
        "en": "With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": {
        "operation": "euclidean_division",
        "inputs": [
          "79",
          "7",
          "11"
        ],
        "output": "2",
        "label": "interval value79 trial divisor7"
      }
    },
    {
      "id": "S100",
      "uses": [
        "G001",
        "S099"
      ],
      "before": "interval value79 trial divisor8:natural dividend=79;positive divisor=8",
      "after": "79=9·8+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": [
          "79",
          "8",
          "9"
        ],
        "output": "7",
        "label": "interval value79 trial divisor8"
      }
    },
    {
      "id": "S101",
      "uses": [
        "G001",
        "S100"
      ],
      "before": "phase=interval value79;tried ascending divisors=[2, 3, 4, 5, 6, 7, 8];sqrt floor=8",
      "after": "least factor at least2=79;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": "S102",
      "uses": [
        "G001",
        "S101"
      ],
      "before": "phase=interval value80;n=80",
      "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": "S103",
      "uses": [
        "G001",
        "S102"
      ],
      "before": "interval value80 trial divisor2:natural dividend=80;positive divisor=2",
      "after": "80=40·2+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": [
          "80",
          "2",
          "40"
        ],
        "output": "0",
        "label": "interval value80 trial divisor2"
      }
    },
    {
      "id": "S104",
      "uses": [
        "G001",
        "S103"
      ],
      "before": "phase=interval value80;tried ascending divisors=[2];sqrt floor=8",
      "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": "S105",
      "uses": [
        "G001",
        "S104"
      ],
      "before": "inclusive interval=[60, 80]",
      "after": "all primes=[61, 67, 71, 73, 79];count=5",
      "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": [
      60,
      80,
      [
        61,
        67,
        71,
        73,
        79
      ],
      5
    ]
  },
  "conclusion": {
    "canonical_value": "(60, 80, [61, 67, 71, 73, 79], 5)",
    "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": 88,
      "original_line_sha256": "a45bff117122fdb36122718998e10f764eceb3d6d49c95cd72fe27532dc4b602",
      "question_compact_sha256": "e7f423eaea0b406bba8f8efda7a9088024223370364c2538bbea3da35c5c573a",
      "english_row_compact_sha256": "98109840af39377c109732c0ba129bef15954e69335815b5506f8571a8498f58",
      "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": "c795922eaeebb51653e468bce9b2aea206cc800121827fd0c5f0de2f8cb12e5f",
    "original_candidate_line_sha256": "c795922eaeebb51653e468bce9b2aea206cc800121827fd0c5f0de2f8cb12e5f"
  },
  "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": 105,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "45147d4c2990a2407e305f8a1c1ce3c80c00d996ab0858588fcd9153c0a1f717",
      "arithmetic_steps": 61
    },
    "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."
  }
}