{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261013-advance-20261018-r04-prime-number-0078",
  "pilot_number": "R0017-09153",
  "number": 9153,
  "family_id": "qf-ext18-prime-interval-inventory",
  "title": {
    "zh-Hant": "第 9153 題：自然數質數與因數",
    "en": "Question 9153: Every Prime in an Inclusive Bounded Interval"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261013-advance-20261018-r04-prime-number-0078",
    "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\":10,\"hi\":30}。請依 [lo,hi,primes,count] 的欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(10, 30, [11, 13, 17, 19, 23, 29], 6)",
      "explanation_zh": "擴展基礎核對=(10, 30, [11, 13, 17, 19, 23, 29], 6)；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": 1,
      "skills": [
        "自然數質數與因數"
      ]
    },
    "variant": {
      "family_id": "qf-ext18-prime-interval-inventory",
      "seed": 202610184078,
      "parameters": {
        "lo": 10,
        "hi": 30,
        "domain": "Nat_prime",
        "operation": "prime-interval-inventory",
        "semantic_template": "qf-ext18-prime-interval-inventory/{\"hi\":30,\"lo\":10}"
      }
    },
    "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-0078",
    "batch_id": "batch-20261013-advance-20261018-r04-prime-number-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "0a0cc84cce809122590e99dbb91a4dd1f3c3739f5a5307905ea1223d7371f8aa",
      "family_id": "qf-ext18-prime-interval-inventory",
      "seed": 202610184078,
      "semantic_template": "qf-ext18-prime-interval-inventory/{\"hi\":30,\"lo\":10}"
    },
    "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=10, hi=30, with lo≤hi and a bounded window. Give (lo, hi, every prime in [lo,hi] in increasing order, count).",
      "answer": {
        "kind": "expression",
        "value": "(10, 30, [11, 13, 17, 19, 23, 29], 6)",
        "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 (10, 30, [11, 13, 17, 19, 23, 29], 6)."
      },
      "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\":10,\"hi\":30}。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=10, hi=30, 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 value10;n=10",
      "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 value10 trial divisor2:natural dividend=10;positive divisor=2",
      "after": "10=5·2+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": [
          "10",
          "2",
          "5"
        ],
        "output": "0",
        "label": "interval value10 trial divisor2"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "phase=interval value10;tried ascending divisors=[2];sqrt floor=3",
      "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 value11;n=11",
      "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 value11 trial divisor2:natural dividend=11;positive divisor=2",
      "after": "11=5·2+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": [
          "11",
          "2",
          "5"
        ],
        "output": "1",
        "label": "interval value11 trial divisor2"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "interval value11 trial divisor3:natural dividend=11;positive divisor=3",
      "after": "11=3·3+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": [
          "11",
          "3",
          "3"
        ],
        "output": "2",
        "label": "interval value11 trial divisor3"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "phase=interval value11;tried ascending divisors=[2, 3];sqrt floor=3",
      "after": "least factor at least2=11;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": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "phase=interval value12;n=12",
      "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": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "interval value12 trial divisor2:natural dividend=12;positive divisor=2",
      "after": "12=6·2+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": [
          "12",
          "2",
          "6"
        ],
        "output": "0",
        "label": "interval value12 trial divisor2"
      }
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "phase=interval value12;tried ascending divisors=[2];sqrt floor=3",
      "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": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "phase=interval value13;n=13",
      "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": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "interval value13 trial divisor2:natural dividend=13;positive divisor=2",
      "after": "13=6·2+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": [
          "13",
          "2",
          "6"
        ],
        "output": "1",
        "label": "interval value13 trial divisor2"
      }
    },
    {
      "id": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "interval value13 trial divisor3:natural dividend=13;positive divisor=3",
      "after": "13=4·3+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": [
          "13",
          "3",
          "4"
        ],
        "output": "1",
        "label": "interval value13 trial divisor3"
      }
    },
    {
      "id": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "phase=interval value13;tried ascending divisors=[2, 3];sqrt floor=3",
      "after": "least factor at least2=13;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": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "phase=interval value14;n=14",
      "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 value14 trial divisor2:natural dividend=14;positive divisor=2",
      "after": "14=7·2+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": [
          "14",
          "2",
          "7"
        ],
        "output": "0",
        "label": "interval value14 trial divisor2"
      }
    },
    {
      "id": "S018",
      "uses": [
        "G001",
        "S017"
      ],
      "before": "phase=interval value14;tried ascending divisors=[2];sqrt floor=3",
      "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": "S019",
      "uses": [
        "G001",
        "S018"
      ],
      "before": "phase=interval value15;n=15",
      "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": "S020",
      "uses": [
        "G001",
        "S019"
      ],
      "before": "interval value15 trial divisor2:natural dividend=15;positive divisor=2",
      "after": "15=7·2+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": [
          "15",
          "2",
          "7"
        ],
        "output": "1",
        "label": "interval value15 trial divisor2"
      }
    },
    {
      "id": "S021",
      "uses": [
        "G001",
        "S020"
      ],
      "before": "interval value15 trial divisor3:natural dividend=15;positive divisor=3",
      "after": "15=5·3+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": [
          "15",
          "3",
          "5"
        ],
        "output": "0",
        "label": "interval value15 trial divisor3"
      }
    },
    {
      "id": "S022",
      "uses": [
        "G001",
        "S021"
      ],
      "before": "phase=interval value15;tried ascending divisors=[2, 3];sqrt floor=3",
      "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": "S023",
      "uses": [
        "G001",
        "S022"
      ],
      "before": "phase=interval value16;n=16",
      "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 value16 trial divisor2:natural dividend=16;positive divisor=2",
      "after": "16=8·2+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": [
          "16",
          "2",
          "8"
        ],
        "output": "0",
        "label": "interval value16 trial divisor2"
      }
    },
    {
      "id": "S025",
      "uses": [
        "G001",
        "S024"
      ],
      "before": "phase=interval value16;tried ascending divisors=[2];sqrt floor=4",
      "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": "S026",
      "uses": [
        "G001",
        "S025"
      ],
      "before": "phase=interval value17;n=17",
      "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": "S027",
      "uses": [
        "G001",
        "S026"
      ],
      "before": "interval value17 trial divisor2:natural dividend=17;positive divisor=2",
      "after": "17=8·2+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": [
          "17",
          "2",
          "8"
        ],
        "output": "1",
        "label": "interval value17 trial divisor2"
      }
    },
    {
      "id": "S028",
      "uses": [
        "G001",
        "S027"
      ],
      "before": "interval value17 trial divisor3:natural dividend=17;positive divisor=3",
      "after": "17=5·3+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": [
          "17",
          "3",
          "5"
        ],
        "output": "2",
        "label": "interval value17 trial divisor3"
      }
    },
    {
      "id": "S029",
      "uses": [
        "G001",
        "S028"
      ],
      "before": "interval value17 trial divisor4:natural dividend=17;positive divisor=4",
      "after": "17=4·4+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": [
          "17",
          "4",
          "4"
        ],
        "output": "1",
        "label": "interval value17 trial divisor4"
      }
    },
    {
      "id": "S030",
      "uses": [
        "G001",
        "S029"
      ],
      "before": "phase=interval value17;tried ascending divisors=[2, 3, 4];sqrt floor=4",
      "after": "least factor at least2=17;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": "S031",
      "uses": [
        "G001",
        "S030"
      ],
      "before": "phase=interval value18;n=18",
      "after": "n>=2=是",
      "relation": "implies",
      "rule_id": "prime_lower_bound_guard",
      "conditions": [],
      "explanation": {
        "zh-Hant": "質數必須至少2；0與1先排除，不能把沒有非平凡試除因子誤當質數。",
        "en": "A prime must be at least2.Exclude0 and1 first rather than treating an absence of trial factors as primality."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S032",
      "uses": [
        "G001",
        "S031"
      ],
      "before": "interval value18 trial divisor2:natural dividend=18;positive divisor=2",
      "after": "18=9·2+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": [
          "18",
          "2",
          "9"
        ],
        "output": "0",
        "label": "interval value18 trial divisor2"
      }
    },
    {
      "id": "S033",
      "uses": [
        "G001",
        "S032"
      ],
      "before": "phase=interval value18;tried ascending divisors=[2];sqrt floor=4",
      "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": "S034",
      "uses": [
        "G001",
        "S033"
      ],
      "before": "phase=interval value19;n=19",
      "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": "S035",
      "uses": [
        "G001",
        "S034"
      ],
      "before": "interval value19 trial divisor2:natural dividend=19;positive divisor=2",
      "after": "19=9·2+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": [
          "19",
          "2",
          "9"
        ],
        "output": "1",
        "label": "interval value19 trial divisor2"
      }
    },
    {
      "id": "S036",
      "uses": [
        "G001",
        "S035"
      ],
      "before": "interval value19 trial divisor3:natural dividend=19;positive divisor=3",
      "after": "19=6·3+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": [
          "19",
          "3",
          "6"
        ],
        "output": "1",
        "label": "interval value19 trial divisor3"
      }
    },
    {
      "id": "S037",
      "uses": [
        "G001",
        "S036"
      ],
      "before": "interval value19 trial divisor4:natural dividend=19;positive divisor=4",
      "after": "19=4·4+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": [
          "19",
          "4",
          "4"
        ],
        "output": "3",
        "label": "interval value19 trial divisor4"
      }
    },
    {
      "id": "S038",
      "uses": [
        "G001",
        "S037"
      ],
      "before": "phase=interval value19;tried ascending divisors=[2, 3, 4];sqrt floor=4",
      "after": "least factor at least2=19;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": "S039",
      "uses": [
        "G001",
        "S038"
      ],
      "before": "phase=interval value20;n=20",
      "after": "n>=2=是",
      "relation": "implies",
      "rule_id": "prime_lower_bound_guard",
      "conditions": [],
      "explanation": {
        "zh-Hant": "質數必須至少2；0與1先排除，不能把沒有非平凡試除因子誤當質數。",
        "en": "A prime must be at least2.Exclude0 and1 first rather than treating an absence of trial factors as primality."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S040",
      "uses": [
        "G001",
        "S039"
      ],
      "before": "interval value20 trial divisor2:natural dividend=20;positive divisor=2",
      "after": "20=10·2+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": [
          "20",
          "2",
          "10"
        ],
        "output": "0",
        "label": "interval value20 trial divisor2"
      }
    },
    {
      "id": "S041",
      "uses": [
        "G001",
        "S040"
      ],
      "before": "phase=interval value20;tried ascending divisors=[2];sqrt floor=4",
      "after": "least factor at least2=2;prime=否",
      "relation": "implies",
      "rule_id": "complete_small_factor_bound",
      "conditions": [],
      "explanation": {
        "zh-Hant": "若n=uv合數且u,v≥2，至少一因子≤sqrt(n)。依序找到首個零餘數就是最小因子；完整界內皆無因子時n自己是最小至少2因子，故為質數。",
        "en": "If n=uv is composite with u,v≥2,one factor is≤sqrt(n).The first zero remainder in ascending order gives the least factor;if none exists through the full bound,n itself is the least factor≥2 and is prime."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S042",
      "uses": [
        "G001",
        "S041"
      ],
      "before": "phase=interval value21;n=21",
      "after": "n>=2=是",
      "relation": "implies",
      "rule_id": "prime_lower_bound_guard",
      "conditions": [],
      "explanation": {
        "zh-Hant": "質數必須至少2；0與1先排除，不能把沒有非平凡試除因子誤當質數。",
        "en": "A prime must be at least2.Exclude0 and1 first rather than treating an absence of trial factors as primality."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S043",
      "uses": [
        "G001",
        "S042"
      ],
      "before": "interval value21 trial divisor2:natural dividend=21;positive divisor=2",
      "after": "21=10·2+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": [
          "21",
          "2",
          "10"
        ],
        "output": "1",
        "label": "interval value21 trial divisor2"
      }
    },
    {
      "id": "S044",
      "uses": [
        "G001",
        "S043"
      ],
      "before": "interval value21 trial divisor3:natural dividend=21;positive divisor=3",
      "after": "21=7·3+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": [
          "21",
          "3",
          "7"
        ],
        "output": "0",
        "label": "interval value21 trial divisor3"
      }
    },
    {
      "id": "S045",
      "uses": [
        "G001",
        "S044"
      ],
      "before": "phase=interval value21;tried ascending divisors=[2, 3];sqrt floor=4",
      "after": "least factor at least2=3;prime=否",
      "relation": "implies",
      "rule_id": "complete_small_factor_bound",
      "conditions": [],
      "explanation": {
        "zh-Hant": "若n=uv合數且u,v≥2，至少一因子≤sqrt(n)。依序找到首個零餘數就是最小因子；完整界內皆無因子時n自己是最小至少2因子，故為質數。",
        "en": "If n=uv is composite with u,v≥2,one factor is≤sqrt(n).The first zero remainder in ascending order gives the least factor;if none exists through the full bound,n itself is the least factor≥2 and is prime."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S046",
      "uses": [
        "G001",
        "S045"
      ],
      "before": "phase=interval value22;n=22",
      "after": "n>=2=是",
      "relation": "implies",
      "rule_id": "prime_lower_bound_guard",
      "conditions": [],
      "explanation": {
        "zh-Hant": "質數必須至少2；0與1先排除，不能把沒有非平凡試除因子誤當質數。",
        "en": "A prime must be at least2.Exclude0 and1 first rather than treating an absence of trial factors as primality."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S047",
      "uses": [
        "G001",
        "S046"
      ],
      "before": "interval value22 trial divisor2:natural dividend=22;positive divisor=2",
      "after": "22=11·2+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": [
          "22",
          "2",
          "11"
        ],
        "output": "0",
        "label": "interval value22 trial divisor2"
      }
    },
    {
      "id": "S048",
      "uses": [
        "G001",
        "S047"
      ],
      "before": "phase=interval value22;tried ascending divisors=[2];sqrt floor=4",
      "after": "least factor at least2=2;prime=否",
      "relation": "implies",
      "rule_id": "complete_small_factor_bound",
      "conditions": [],
      "explanation": {
        "zh-Hant": "若n=uv合數且u,v≥2，至少一因子≤sqrt(n)。依序找到首個零餘數就是最小因子；完整界內皆無因子時n自己是最小至少2因子，故為質數。",
        "en": "If n=uv is composite with u,v≥2,one factor is≤sqrt(n).The first zero remainder in ascending order gives the least factor;if none exists through the full bound,n itself is the least factor≥2 and is prime."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S049",
      "uses": [
        "G001",
        "S048"
      ],
      "before": "phase=interval value23;n=23",
      "after": "n>=2=是",
      "relation": "implies",
      "rule_id": "prime_lower_bound_guard",
      "conditions": [],
      "explanation": {
        "zh-Hant": "質數必須至少2；0與1先排除，不能把沒有非平凡試除因子誤當質數。",
        "en": "A prime must be at least2.Exclude0 and1 first rather than treating an absence of trial factors as primality."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S050",
      "uses": [
        "G001",
        "S049"
      ],
      "before": "interval value23 trial divisor2:natural dividend=23;positive divisor=2",
      "after": "23=11·2+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": [
          "23",
          "2",
          "11"
        ],
        "output": "1",
        "label": "interval value23 trial divisor2"
      }
    },
    {
      "id": "S051",
      "uses": [
        "G001",
        "S050"
      ],
      "before": "interval value23 trial divisor3:natural dividend=23;positive divisor=3",
      "after": "23=7·3+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": [
          "23",
          "3",
          "7"
        ],
        "output": "2",
        "label": "interval value23 trial divisor3"
      }
    },
    {
      "id": "S052",
      "uses": [
        "G001",
        "S051"
      ],
      "before": "interval value23 trial divisor4:natural dividend=23;positive divisor=4",
      "after": "23=5·4+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": [
          "23",
          "4",
          "5"
        ],
        "output": "3",
        "label": "interval value23 trial divisor4"
      }
    },
    {
      "id": "S053",
      "uses": [
        "G001",
        "S052"
      ],
      "before": "phase=interval value23;tried ascending divisors=[2, 3, 4];sqrt floor=4",
      "after": "least factor at least2=23;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": "S054",
      "uses": [
        "G001",
        "S053"
      ],
      "before": "phase=interval value24;n=24",
      "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": "S055",
      "uses": [
        "G001",
        "S054"
      ],
      "before": "interval value24 trial divisor2:natural dividend=24;positive divisor=2",
      "after": "24=12·2+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": [
          "24",
          "2",
          "12"
        ],
        "output": "0",
        "label": "interval value24 trial divisor2"
      }
    },
    {
      "id": "S056",
      "uses": [
        "G001",
        "S055"
      ],
      "before": "phase=interval value24;tried ascending divisors=[2];sqrt floor=4",
      "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": "S057",
      "uses": [
        "G001",
        "S056"
      ],
      "before": "phase=interval value25;n=25",
      "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": "S058",
      "uses": [
        "G001",
        "S057"
      ],
      "before": "interval value25 trial divisor2:natural dividend=25;positive divisor=2",
      "after": "25=12·2+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": [
          "25",
          "2",
          "12"
        ],
        "output": "1",
        "label": "interval value25 trial divisor2"
      }
    },
    {
      "id": "S059",
      "uses": [
        "G001",
        "S058"
      ],
      "before": "interval value25 trial divisor3:natural dividend=25;positive divisor=3",
      "after": "25=8·3+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": [
          "25",
          "3",
          "8"
        ],
        "output": "1",
        "label": "interval value25 trial divisor3"
      }
    },
    {
      "id": "S060",
      "uses": [
        "G001",
        "S059"
      ],
      "before": "interval value25 trial divisor4:natural dividend=25;positive divisor=4",
      "after": "25=6·4+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": [
          "25",
          "4",
          "6"
        ],
        "output": "1",
        "label": "interval value25 trial divisor4"
      }
    },
    {
      "id": "S061",
      "uses": [
        "G001",
        "S060"
      ],
      "before": "interval value25 trial divisor5:natural dividend=25;positive divisor=5",
      "after": "25=5·5+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": [
          "25",
          "5",
          "5"
        ],
        "output": "0",
        "label": "interval value25 trial divisor5"
      }
    },
    {
      "id": "S062",
      "uses": [
        "G001",
        "S061"
      ],
      "before": "phase=interval value25;tried ascending divisors=[2, 3, 4, 5];sqrt floor=5",
      "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": "S063",
      "uses": [
        "G001",
        "S062"
      ],
      "before": "phase=interval value26;n=26",
      "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 value26 trial divisor2:natural dividend=26;positive divisor=2",
      "after": "26=13·2+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": [
          "26",
          "2",
          "13"
        ],
        "output": "0",
        "label": "interval value26 trial divisor2"
      }
    },
    {
      "id": "S065",
      "uses": [
        "G001",
        "S064"
      ],
      "before": "phase=interval value26;tried ascending divisors=[2];sqrt floor=5",
      "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": "S066",
      "uses": [
        "G001",
        "S065"
      ],
      "before": "phase=interval value27;n=27",
      "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": "S067",
      "uses": [
        "G001",
        "S066"
      ],
      "before": "interval value27 trial divisor2:natural dividend=27;positive divisor=2",
      "after": "27=13·2+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": [
          "27",
          "2",
          "13"
        ],
        "output": "1",
        "label": "interval value27 trial divisor2"
      }
    },
    {
      "id": "S068",
      "uses": [
        "G001",
        "S067"
      ],
      "before": "interval value27 trial divisor3:natural dividend=27;positive divisor=3",
      "after": "27=9·3+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": [
          "27",
          "3",
          "9"
        ],
        "output": "0",
        "label": "interval value27 trial divisor3"
      }
    },
    {
      "id": "S069",
      "uses": [
        "G001",
        "S068"
      ],
      "before": "phase=interval value27;tried ascending divisors=[2, 3];sqrt floor=5",
      "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": "S070",
      "uses": [
        "G001",
        "S069"
      ],
      "before": "phase=interval value28;n=28",
      "after": "n>=2=是",
      "relation": "implies",
      "rule_id": "prime_lower_bound_guard",
      "conditions": [],
      "explanation": {
        "zh-Hant": "質數必須至少2；0與1先排除，不能把沒有非平凡試除因子誤當質數。",
        "en": "A prime must be at least2.Exclude0 and1 first rather than treating an absence of trial factors as primality."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S071",
      "uses": [
        "G001",
        "S070"
      ],
      "before": "interval value28 trial divisor2:natural dividend=28;positive divisor=2",
      "after": "28=14·2+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": [
          "28",
          "2",
          "14"
        ],
        "output": "0",
        "label": "interval value28 trial divisor2"
      }
    },
    {
      "id": "S072",
      "uses": [
        "G001",
        "S071"
      ],
      "before": "phase=interval value28;tried ascending divisors=[2];sqrt floor=5",
      "after": "least factor at least2=2;prime=否",
      "relation": "implies",
      "rule_id": "complete_small_factor_bound",
      "conditions": [],
      "explanation": {
        "zh-Hant": "若n=uv合數且u,v≥2，至少一因子≤sqrt(n)。依序找到首個零餘數就是最小因子；完整界內皆無因子時n自己是最小至少2因子，故為質數。",
        "en": "If n=uv is composite with u,v≥2,one factor is≤sqrt(n).The first zero remainder in ascending order gives the least factor;if none exists through the full bound,n itself is the least factor≥2 and is prime."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S073",
      "uses": [
        "G001",
        "S072"
      ],
      "before": "phase=interval value29;n=29",
      "after": "n>=2=是",
      "relation": "implies",
      "rule_id": "prime_lower_bound_guard",
      "conditions": [],
      "explanation": {
        "zh-Hant": "質數必須至少2；0與1先排除，不能把沒有非平凡試除因子誤當質數。",
        "en": "A prime must be at least2.Exclude0 and1 first rather than treating an absence of trial factors as primality."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S074",
      "uses": [
        "G001",
        "S073"
      ],
      "before": "interval value29 trial divisor2:natural dividend=29;positive divisor=2",
      "after": "29=14·2+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": [
          "29",
          "2",
          "14"
        ],
        "output": "1",
        "label": "interval value29 trial divisor2"
      }
    },
    {
      "id": "S075",
      "uses": [
        "G001",
        "S074"
      ],
      "before": "interval value29 trial divisor3:natural dividend=29;positive divisor=3",
      "after": "29=9·3+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": [
          "29",
          "3",
          "9"
        ],
        "output": "2",
        "label": "interval value29 trial divisor3"
      }
    },
    {
      "id": "S076",
      "uses": [
        "G001",
        "S075"
      ],
      "before": "interval value29 trial divisor4:natural dividend=29;positive divisor=4",
      "after": "29=7·4+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": [
          "29",
          "4",
          "7"
        ],
        "output": "1",
        "label": "interval value29 trial divisor4"
      }
    },
    {
      "id": "S077",
      "uses": [
        "G001",
        "S076"
      ],
      "before": "interval value29 trial divisor5:natural dividend=29;positive divisor=5",
      "after": "29=5·5+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": [
          "29",
          "5",
          "5"
        ],
        "output": "4",
        "label": "interval value29 trial divisor5"
      }
    },
    {
      "id": "S078",
      "uses": [
        "G001",
        "S077"
      ],
      "before": "phase=interval value29;tried ascending divisors=[2, 3, 4, 5];sqrt floor=5",
      "after": "least factor at least2=29;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": "S079",
      "uses": [
        "G001",
        "S078"
      ],
      "before": "phase=interval value30;n=30",
      "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 value30 trial divisor2:natural dividend=30;positive divisor=2",
      "after": "30=15·2+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": [
          "30",
          "2",
          "15"
        ],
        "output": "0",
        "label": "interval value30 trial divisor2"
      }
    },
    {
      "id": "S081",
      "uses": [
        "G001",
        "S080"
      ],
      "before": "phase=interval value30;tried ascending divisors=[2];sqrt floor=5",
      "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": "inclusive interval=[10, 30]",
      "after": "all primes=[11, 13, 17, 19, 23, 29];count=6",
      "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": [
      10,
      30,
      [
        11,
        13,
        17,
        19,
        23,
        29
      ],
      6
    ]
  },
  "conclusion": {
    "canonical_value": "(10, 30, [11, 13, 17, 19, 23, 29], 6)",
    "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": 78,
      "original_line_sha256": "a6f122631c04022323f4b44d64d1a5ece3fa1b0543af897227d939b15326c568",
      "question_compact_sha256": "0a0cc84cce809122590e99dbb91a4dd1f3c3739f5a5307905ea1223d7371f8aa",
      "english_row_compact_sha256": "1378f4a17274d57e01af127c1c91a9074d861de8b235b0c80cb983124278479b",
      "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": "00d33b6453899856e6bd033a15ee8a8460e22669fe5f2d9071d793d2f4ef0d26",
    "original_candidate_line_sha256": "00d33b6453899856e6bd033a15ee8a8460e22669fe5f2d9071d793d2f4ef0d26"
  },
  "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": 82,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "f26aed986acdab15bcaf7068f8c6aff343f2791b2b4b8e82511dce7c1b32cb39",
      "arithmetic_steps": 38
    },
    "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."
  }
}