{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261013-advance-20261018-r02-natural-divisibility-0046",
  "pilot_number": "R0016-08821",
  "number": 8821,
  "family_id": "qf-ext18-divisibility-divisor-set",
  "title": {
    "zh-Hant": "第 8821 題：自然數整除與零邊界",
    "en": "Question 8821: Complete Positive Divisor Inventory Including the Zero Case"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261013-advance-20261018-r02-natural-divisibility-0046",
    "batch_id": "batch-20261013-advance-20261018-r02-natural-divisibility-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-natural-divisibility"
    ],
    "title_zh": "自然數整除與零邊界",
    "stem_zh": "自然數包含0。操作=divisibility-divisor-set；資料={\"b\":30}。請依 [b,positive_divisors_or_all,count_or_無限] 的欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(30, [1, 2, 3, 5, 6, 10, 15, 30], 8)",
      "explanation_zh": "擴展基礎核對=(30, [1, 2, 3, 5, 6, 10, 15, 30], 8)；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∣b當且僅當b=0；a∣0對所有自然數a成立。"
    },
    "difficulty": {
      "band": "upper_secondary",
      "level": 3,
      "skills": [
        "自然數整除與零邊界"
      ]
    },
    "variant": {
      "family_id": "qf-ext18-divisibility-divisor-set",
      "seed": 202610182046,
      "parameters": {
        "b": 30,
        "domain": "Nat_divisibility",
        "operation": "divisibility-divisor-set",
        "semantic_template": "qf-ext18-divisibility-divisor-set/{\"b\":30}"
      }
    },
    "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-r02-natural-divisibility-0046",
    "batch_id": "batch-20261013-advance-20261018-r02-natural-divisibility-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "7d61c0446c07279e818853d34cef67aaeb1ece20f6db90830085da19999e4b5c",
      "family_id": "qf-ext18-divisibility-divisor-set",
      "seed": 202610182046,
      "semantic_template": "qf-ext18-divisibility-divisor-set/{\"b\":30}"
    },
    "fields": {
      "title": "Complete Positive Divisor Inventory Including the Zero Case",
      "stem": "Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. b=30. Give (b, all positive divisors in increasing order or all-positive-naturals, count or infinite). This inventory excludes divisor 0 even when b=0.",
      "answer": {
        "kind": "expression",
        "value": "(30, [1, 2, 3, 5, 6, 10, 15, 30], 8)",
        "explanation": "For a positive input enumerate every positive divisor; every positive natural divides zero, so its positive-divisor collection is infinite. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (30, [1, 2, 3, 5, 6, 10, 15, 30], 8)."
      },
      "skills": [
        "Complete Positive Divisor Inventory Including the Zero Case"
      ],
      "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": "自然數包含0。操作=divisibility-divisor-set；資料={\"b\":30}。請依 [b,positive_divisors_or_all,count_or_無限] 的欄序回答。",
        "en": "Natural numbers include 0. Every a divides 0; 0 divides b exactly when b=0. Divisibility witnesses are natural-number multipliers, not approximate quotients. b=30. Give (b, all positive divisors in increasing order or all-positive-naturals, count or infinite). This inventory excludes divisor 0 even when b=0."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "natural numbers include0;b is divisible by a iff b=a*k for some natural k",
      "after": "0 divides only0;every natural divides0",
      "relation": "implies",
      "rule_id": "natural_divisibility_zero_scope",
      "conditions": [],
      "explanation": {
        "zh-Hant": "整除是存在乘數的關係，不先假定除數正或非零；a∣0包含a=0。正因數清單另排除0，兩個問題不混用。",
        "en": "Divisibility is an existential multiplier relation without presuming a positive or nonzero divisor.Every natural,including0,divides0.A positive-divisor inventory separately excludes0."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "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": "positive divisor candidate1:divide natural 30 by positive 1",
      "after": "30=30·1+0",
      "relation": "implies",
      "rule_id": "natural_euclidean_division_witness",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數給唯一商餘，餘數滿足0≤r<divisor；這個完整等式同時核商與餘數，不作除以0。",
        "en": "A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "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",
          "1",
          "30"
        ],
        "output": "0",
        "label": "positive divisor candidate1"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "positive divisor candidate1:a=1;b=30;quotient=30;remainder=0",
      "after": "divides=是;witness/status=30",
      "relation": "implies",
      "rule_id": "positive_divisor_remainder_criterion",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正a下，餘數0時商就是自然乘數見證；非零餘數由商餘唯一性排除整除，不把截斷商當見證。",
        "en": "For positive a,a zero remainder makes the quotient a natural multiplier witness.A nonzero remainder rules out divisibility by uniqueness;the truncated quotient is not a witness."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "positive divisor candidate2:divide natural 30 by positive 2",
      "after": "30=15·2+0",
      "relation": "implies",
      "rule_id": "natural_euclidean_division_witness",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數給唯一商餘，餘數滿足0≤r<divisor；這個完整等式同時核商與餘數，不作除以0。",
        "en": "A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "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": "positive divisor candidate2"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "positive divisor candidate2:a=2;b=30;quotient=15;remainder=0",
      "after": "divides=是;witness/status=15",
      "relation": "implies",
      "rule_id": "positive_divisor_remainder_criterion",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正a下，餘數0時商就是自然乘數見證；非零餘數由商餘唯一性排除整除，不把截斷商當見證。",
        "en": "For positive a,a zero remainder makes the quotient a natural multiplier witness.A nonzero remainder rules out divisibility by uniqueness;the truncated quotient is not a witness."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "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": "positive divisor candidate3:divide natural 30 by positive 3",
      "after": "30=10·3+0",
      "relation": "implies",
      "rule_id": "natural_euclidean_division_witness",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數給唯一商餘，餘數滿足0≤r<divisor；這個完整等式同時核商與餘數，不作除以0。",
        "en": "A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "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",
          "3",
          "10"
        ],
        "output": "0",
        "label": "positive divisor candidate3"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "positive divisor candidate3:a=3;b=30;quotient=10;remainder=0",
      "after": "divides=是;witness/status=10",
      "relation": "implies",
      "rule_id": "positive_divisor_remainder_criterion",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正a下，餘數0時商就是自然乘數見證；非零餘數由商餘唯一性排除整除，不把截斷商當見證。",
        "en": "For positive a,a zero remainder makes the quotient a natural multiplier witness.A nonzero remainder rules out divisibility by uniqueness;the truncated quotient is not a witness."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "positive divisor candidate4:divide natural 30 by positive 4",
      "after": "30=7·4+2",
      "relation": "implies",
      "rule_id": "natural_euclidean_division_witness",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數給唯一商餘，餘數滿足0≤r<divisor；這個完整等式同時核商與餘數，不作除以0。",
        "en": "A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "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",
          "4",
          "7"
        ],
        "output": "2",
        "label": "positive divisor candidate4"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "positive divisor candidate4:a=4;b=30;quotient=7;remainder=2",
      "after": "divides=否;witness/status=不存在",
      "relation": "implies",
      "rule_id": "positive_divisor_remainder_criterion",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正a下，餘數0時商就是自然乘數見證；非零餘數由商餘唯一性排除整除，不把截斷商當見證。",
        "en": "For positive a,a zero remainder makes the quotient a natural multiplier witness.A nonzero remainder rules out divisibility by uniqueness;the truncated quotient is not a witness."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "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": "positive divisor candidate5:divide natural 30 by positive 5",
      "after": "30=6·5+0",
      "relation": "implies",
      "rule_id": "natural_euclidean_division_witness",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數給唯一商餘，餘數滿足0≤r<divisor；這個完整等式同時核商與餘數，不作除以0。",
        "en": "A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "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",
          "5",
          "6"
        ],
        "output": "0",
        "label": "positive divisor candidate5"
      }
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "positive divisor candidate5:a=5;b=30;quotient=6;remainder=0",
      "after": "divides=是;witness/status=6",
      "relation": "implies",
      "rule_id": "positive_divisor_remainder_criterion",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正a下，餘數0時商就是自然乘數見證；非零餘數由商餘唯一性排除整除，不把截斷商當見證。",
        "en": "For positive a,a zero remainder makes the quotient a natural multiplier witness.A nonzero remainder rules out divisibility by uniqueness;the truncated quotient is not a witness."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "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": "positive divisor candidate6:divide natural 30 by positive 6",
      "after": "30=5·6+0",
      "relation": "implies",
      "rule_id": "natural_euclidean_division_witness",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數給唯一商餘，餘數滿足0≤r<divisor；這個完整等式同時核商與餘數，不作除以0。",
        "en": "A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "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",
          "6",
          "5"
        ],
        "output": "0",
        "label": "positive divisor candidate6"
      }
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "positive divisor candidate6:a=6;b=30;quotient=5;remainder=0",
      "after": "divides=是;witness/status=5",
      "relation": "implies",
      "rule_id": "positive_divisor_remainder_criterion",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正a下，餘數0時商就是自然乘數見證；非零餘數由商餘唯一性排除整除，不把截斷商當見證。",
        "en": "For positive a,a zero remainder makes the quotient a natural multiplier witness.A nonzero remainder rules out divisibility by uniqueness;the truncated quotient is not a witness."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "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": "positive divisor candidate7:divide natural 30 by positive 7",
      "after": "30=4·7+2",
      "relation": "implies",
      "rule_id": "natural_euclidean_division_witness",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數給唯一商餘，餘數滿足0≤r<divisor；這個完整等式同時核商與餘數，不作除以0。",
        "en": "A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "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",
          "7",
          "4"
        ],
        "output": "2",
        "label": "positive divisor candidate7"
      }
    },
    {
      "id": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "positive divisor candidate7:a=7;b=30;quotient=4;remainder=2",
      "after": "divides=否;witness/status=不存在",
      "relation": "implies",
      "rule_id": "positive_divisor_remainder_criterion",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正a下，餘數0時商就是自然乘數見證；非零餘數由商餘唯一性排除整除，不把截斷商當見證。",
        "en": "For positive a,a zero remainder makes the quotient a natural multiplier witness.A nonzero remainder rules out divisibility by uniqueness;the truncated quotient is not a witness."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "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": "positive divisor candidate8:divide natural 30 by positive 8",
      "after": "30=3·8+6",
      "relation": "implies",
      "rule_id": "natural_euclidean_division_witness",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數給唯一商餘，餘數滿足0≤r<divisor；這個完整等式同時核商與餘數，不作除以0。",
        "en": "A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "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",
          "8",
          "3"
        ],
        "output": "6",
        "label": "positive divisor candidate8"
      }
    },
    {
      "id": "S017",
      "uses": [
        "G001",
        "S016"
      ],
      "before": "positive divisor candidate8:a=8;b=30;quotient=3;remainder=6",
      "after": "divides=否;witness/status=不存在",
      "relation": "implies",
      "rule_id": "positive_divisor_remainder_criterion",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正a下，餘數0時商就是自然乘數見證；非零餘數由商餘唯一性排除整除，不把截斷商當見證。",
        "en": "For positive a,a zero remainder makes the quotient a natural multiplier witness.A nonzero remainder rules out divisibility by uniqueness;the truncated quotient is not a witness."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S018",
      "uses": [
        "G001",
        "S017"
      ],
      "before": "positive divisor candidate9:divide natural 30 by positive 9",
      "after": "30=3·9+3",
      "relation": "implies",
      "rule_id": "natural_euclidean_division_witness",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數給唯一商餘，餘數滿足0≤r<divisor；這個完整等式同時核商與餘數，不作除以0。",
        "en": "A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "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",
          "9",
          "3"
        ],
        "output": "3",
        "label": "positive divisor candidate9"
      }
    },
    {
      "id": "S019",
      "uses": [
        "G001",
        "S018"
      ],
      "before": "positive divisor candidate9:a=9;b=30;quotient=3;remainder=3",
      "after": "divides=否;witness/status=不存在",
      "relation": "implies",
      "rule_id": "positive_divisor_remainder_criterion",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正a下，餘數0時商就是自然乘數見證；非零餘數由商餘唯一性排除整除，不把截斷商當見證。",
        "en": "For positive a,a zero remainder makes the quotient a natural multiplier witness.A nonzero remainder rules out divisibility by uniqueness;the truncated quotient is not a witness."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "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": "positive divisor candidate10:divide natural 30 by positive 10",
      "after": "30=3·10+0",
      "relation": "implies",
      "rule_id": "natural_euclidean_division_witness",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數給唯一商餘，餘數滿足0≤r<divisor；這個完整等式同時核商與餘數，不作除以0。",
        "en": "A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "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",
          "10",
          "3"
        ],
        "output": "0",
        "label": "positive divisor candidate10"
      }
    },
    {
      "id": "S021",
      "uses": [
        "G001",
        "S020"
      ],
      "before": "positive divisor candidate10:a=10;b=30;quotient=3;remainder=0",
      "after": "divides=是;witness/status=3",
      "relation": "implies",
      "rule_id": "positive_divisor_remainder_criterion",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正a下，餘數0時商就是自然乘數見證；非零餘數由商餘唯一性排除整除，不把截斷商當見證。",
        "en": "For positive a,a zero remainder makes the quotient a natural multiplier witness.A nonzero remainder rules out divisibility by uniqueness;the truncated quotient is not a witness."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S022",
      "uses": [
        "G001",
        "S021"
      ],
      "before": "positive divisor candidate11:divide natural 30 by positive 11",
      "after": "30=2·11+8",
      "relation": "implies",
      "rule_id": "natural_euclidean_division_witness",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數給唯一商餘，餘數滿足0≤r<divisor；這個完整等式同時核商與餘數，不作除以0。",
        "en": "A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "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",
          "11",
          "2"
        ],
        "output": "8",
        "label": "positive divisor candidate11"
      }
    },
    {
      "id": "S023",
      "uses": [
        "G001",
        "S022"
      ],
      "before": "positive divisor candidate11:a=11;b=30;quotient=2;remainder=8",
      "after": "divides=否;witness/status=不存在",
      "relation": "implies",
      "rule_id": "positive_divisor_remainder_criterion",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正a下，餘數0時商就是自然乘數見證；非零餘數由商餘唯一性排除整除，不把截斷商當見證。",
        "en": "For positive a,a zero remainder makes the quotient a natural multiplier witness.A nonzero remainder rules out divisibility by uniqueness;the truncated quotient is not a witness."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "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": "positive divisor candidate12:divide natural 30 by positive 12",
      "after": "30=2·12+6",
      "relation": "implies",
      "rule_id": "natural_euclidean_division_witness",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數給唯一商餘，餘數滿足0≤r<divisor；這個完整等式同時核商與餘數，不作除以0。",
        "en": "A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "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",
          "12",
          "2"
        ],
        "output": "6",
        "label": "positive divisor candidate12"
      }
    },
    {
      "id": "S025",
      "uses": [
        "G001",
        "S024"
      ],
      "before": "positive divisor candidate12:a=12;b=30;quotient=2;remainder=6",
      "after": "divides=否;witness/status=不存在",
      "relation": "implies",
      "rule_id": "positive_divisor_remainder_criterion",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正a下，餘數0時商就是自然乘數見證；非零餘數由商餘唯一性排除整除，不把截斷商當見證。",
        "en": "For positive a,a zero remainder makes the quotient a natural multiplier witness.A nonzero remainder rules out divisibility by uniqueness;the truncated quotient is not a witness."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "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": "positive divisor candidate13:divide natural 30 by positive 13",
      "after": "30=2·13+4",
      "relation": "implies",
      "rule_id": "natural_euclidean_division_witness",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數給唯一商餘，餘數滿足0≤r<divisor；這個完整等式同時核商與餘數，不作除以0。",
        "en": "A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "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",
          "13",
          "2"
        ],
        "output": "4",
        "label": "positive divisor candidate13"
      }
    },
    {
      "id": "S027",
      "uses": [
        "G001",
        "S026"
      ],
      "before": "positive divisor candidate13:a=13;b=30;quotient=2;remainder=4",
      "after": "divides=否;witness/status=不存在",
      "relation": "implies",
      "rule_id": "positive_divisor_remainder_criterion",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正a下，餘數0時商就是自然乘數見證；非零餘數由商餘唯一性排除整除，不把截斷商當見證。",
        "en": "For positive a,a zero remainder makes the quotient a natural multiplier witness.A nonzero remainder rules out divisibility by uniqueness;the truncated quotient is not a witness."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S028",
      "uses": [
        "G001",
        "S027"
      ],
      "before": "positive divisor candidate14:divide natural 30 by positive 14",
      "after": "30=2·14+2",
      "relation": "implies",
      "rule_id": "natural_euclidean_division_witness",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數給唯一商餘，餘數滿足0≤r<divisor；這個完整等式同時核商與餘數，不作除以0。",
        "en": "A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "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",
          "14",
          "2"
        ],
        "output": "2",
        "label": "positive divisor candidate14"
      }
    },
    {
      "id": "S029",
      "uses": [
        "G001",
        "S028"
      ],
      "before": "positive divisor candidate14:a=14;b=30;quotient=2;remainder=2",
      "after": "divides=否;witness/status=不存在",
      "relation": "implies",
      "rule_id": "positive_divisor_remainder_criterion",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正a下，餘數0時商就是自然乘數見證；非零餘數由商餘唯一性排除整除，不把截斷商當見證。",
        "en": "For positive a,a zero remainder makes the quotient a natural multiplier witness.A nonzero remainder rules out divisibility by uniqueness;the truncated quotient is not a witness."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "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": "positive divisor candidate15:divide natural 30 by positive 15",
      "after": "30=2·15+0",
      "relation": "implies",
      "rule_id": "natural_euclidean_division_witness",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數給唯一商餘，餘數滿足0≤r<divisor；這個完整等式同時核商與餘數，不作除以0。",
        "en": "A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "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",
          "15",
          "2"
        ],
        "output": "0",
        "label": "positive divisor candidate15"
      }
    },
    {
      "id": "S031",
      "uses": [
        "G001",
        "S030"
      ],
      "before": "positive divisor candidate15:a=15;b=30;quotient=2;remainder=0",
      "after": "divides=是;witness/status=2",
      "relation": "implies",
      "rule_id": "positive_divisor_remainder_criterion",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正a下，餘數0時商就是自然乘數見證；非零餘數由商餘唯一性排除整除，不把截斷商當見證。",
        "en": "For positive a,a zero remainder makes the quotient a natural multiplier witness.A nonzero remainder rules out divisibility by uniqueness;the truncated quotient is not a witness."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "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": "positive divisor candidate16:divide natural 30 by positive 16",
      "after": "30=1·16+14",
      "relation": "implies",
      "rule_id": "natural_euclidean_division_witness",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數給唯一商餘，餘數滿足0≤r<divisor；這個完整等式同時核商與餘數，不作除以0。",
        "en": "A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "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",
          "16",
          "1"
        ],
        "output": "14",
        "label": "positive divisor candidate16"
      }
    },
    {
      "id": "S033",
      "uses": [
        "G001",
        "S032"
      ],
      "before": "positive divisor candidate16:a=16;b=30;quotient=1;remainder=14",
      "after": "divides=否;witness/status=不存在",
      "relation": "implies",
      "rule_id": "positive_divisor_remainder_criterion",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正a下，餘數0時商就是自然乘數見證；非零餘數由商餘唯一性排除整除，不把截斷商當見證。",
        "en": "For positive a,a zero remainder makes the quotient a natural multiplier witness.A nonzero remainder rules out divisibility by uniqueness;the truncated quotient is not a witness."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "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": "positive divisor candidate17:divide natural 30 by positive 17",
      "after": "30=1·17+13",
      "relation": "implies",
      "rule_id": "natural_euclidean_division_witness",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數給唯一商餘，餘數滿足0≤r<divisor；這個完整等式同時核商與餘數，不作除以0。",
        "en": "A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "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",
          "17",
          "1"
        ],
        "output": "13",
        "label": "positive divisor candidate17"
      }
    },
    {
      "id": "S035",
      "uses": [
        "G001",
        "S034"
      ],
      "before": "positive divisor candidate17:a=17;b=30;quotient=1;remainder=13",
      "after": "divides=否;witness/status=不存在",
      "relation": "implies",
      "rule_id": "positive_divisor_remainder_criterion",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正a下，餘數0時商就是自然乘數見證；非零餘數由商餘唯一性排除整除，不把截斷商當見證。",
        "en": "For positive a,a zero remainder makes the quotient a natural multiplier witness.A nonzero remainder rules out divisibility by uniqueness;the truncated quotient is not a witness."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S036",
      "uses": [
        "G001",
        "S035"
      ],
      "before": "positive divisor candidate18:divide natural 30 by positive 18",
      "after": "30=1·18+12",
      "relation": "implies",
      "rule_id": "natural_euclidean_division_witness",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數給唯一商餘，餘數滿足0≤r<divisor；這個完整等式同時核商與餘數，不作除以0。",
        "en": "A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "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",
          "18",
          "1"
        ],
        "output": "12",
        "label": "positive divisor candidate18"
      }
    },
    {
      "id": "S037",
      "uses": [
        "G001",
        "S036"
      ],
      "before": "positive divisor candidate18:a=18;b=30;quotient=1;remainder=12",
      "after": "divides=否;witness/status=不存在",
      "relation": "implies",
      "rule_id": "positive_divisor_remainder_criterion",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正a下，餘數0時商就是自然乘數見證；非零餘數由商餘唯一性排除整除，不把截斷商當見證。",
        "en": "For positive a,a zero remainder makes the quotient a natural multiplier witness.A nonzero remainder rules out divisibility by uniqueness;the truncated quotient is not a witness."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S038",
      "uses": [
        "G001",
        "S037"
      ],
      "before": "positive divisor candidate19:divide natural 30 by positive 19",
      "after": "30=1·19+11",
      "relation": "implies",
      "rule_id": "natural_euclidean_division_witness",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數給唯一商餘，餘數滿足0≤r<divisor；這個完整等式同時核商與餘數，不作除以0。",
        "en": "A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "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",
          "19",
          "1"
        ],
        "output": "11",
        "label": "positive divisor candidate19"
      }
    },
    {
      "id": "S039",
      "uses": [
        "G001",
        "S038"
      ],
      "before": "positive divisor candidate19:a=19;b=30;quotient=1;remainder=11",
      "after": "divides=否;witness/status=不存在",
      "relation": "implies",
      "rule_id": "positive_divisor_remainder_criterion",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正a下，餘數0時商就是自然乘數見證；非零餘數由商餘唯一性排除整除，不把截斷商當見證。",
        "en": "For positive a,a zero remainder makes the quotient a natural multiplier witness.A nonzero remainder rules out divisibility by uniqueness;the truncated quotient is not a witness."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "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": "positive divisor candidate20:divide natural 30 by positive 20",
      "after": "30=1·20+10",
      "relation": "implies",
      "rule_id": "natural_euclidean_division_witness",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數給唯一商餘，餘數滿足0≤r<divisor；這個完整等式同時核商與餘數，不作除以0。",
        "en": "A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "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",
          "20",
          "1"
        ],
        "output": "10",
        "label": "positive divisor candidate20"
      }
    },
    {
      "id": "S041",
      "uses": [
        "G001",
        "S040"
      ],
      "before": "positive divisor candidate20:a=20;b=30;quotient=1;remainder=10",
      "after": "divides=否;witness/status=不存在",
      "relation": "implies",
      "rule_id": "positive_divisor_remainder_criterion",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正a下，餘數0時商就是自然乘數見證；非零餘數由商餘唯一性排除整除，不把截斷商當見證。",
        "en": "For positive a,a zero remainder makes the quotient a natural multiplier witness.A nonzero remainder rules out divisibility by uniqueness;the truncated quotient is not a witness."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "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": "positive divisor candidate21:divide natural 30 by positive 21",
      "after": "30=1·21+9",
      "relation": "implies",
      "rule_id": "natural_euclidean_division_witness",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數給唯一商餘，餘數滿足0≤r<divisor；這個完整等式同時核商與餘數，不作除以0。",
        "en": "A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "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",
          "21",
          "1"
        ],
        "output": "9",
        "label": "positive divisor candidate21"
      }
    },
    {
      "id": "S043",
      "uses": [
        "G001",
        "S042"
      ],
      "before": "positive divisor candidate21:a=21;b=30;quotient=1;remainder=9",
      "after": "divides=否;witness/status=不存在",
      "relation": "implies",
      "rule_id": "positive_divisor_remainder_criterion",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正a下，餘數0時商就是自然乘數見證；非零餘數由商餘唯一性排除整除，不把截斷商當見證。",
        "en": "For positive a,a zero remainder makes the quotient a natural multiplier witness.A nonzero remainder rules out divisibility by uniqueness;the truncated quotient is not a witness."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "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": "positive divisor candidate22:divide natural 30 by positive 22",
      "after": "30=1·22+8",
      "relation": "implies",
      "rule_id": "natural_euclidean_division_witness",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數給唯一商餘，餘數滿足0≤r<divisor；這個完整等式同時核商與餘數，不作除以0。",
        "en": "A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "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",
          "22",
          "1"
        ],
        "output": "8",
        "label": "positive divisor candidate22"
      }
    },
    {
      "id": "S045",
      "uses": [
        "G001",
        "S044"
      ],
      "before": "positive divisor candidate22:a=22;b=30;quotient=1;remainder=8",
      "after": "divides=否;witness/status=不存在",
      "relation": "implies",
      "rule_id": "positive_divisor_remainder_criterion",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正a下，餘數0時商就是自然乘數見證；非零餘數由商餘唯一性排除整除，不把截斷商當見證。",
        "en": "For positive a,a zero remainder makes the quotient a natural multiplier witness.A nonzero remainder rules out divisibility by uniqueness;the truncated quotient is not a witness."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "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": "positive divisor candidate23:divide natural 30 by positive 23",
      "after": "30=1·23+7",
      "relation": "implies",
      "rule_id": "natural_euclidean_division_witness",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數給唯一商餘，餘數滿足0≤r<divisor；這個完整等式同時核商與餘數，不作除以0。",
        "en": "A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "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",
          "23",
          "1"
        ],
        "output": "7",
        "label": "positive divisor candidate23"
      }
    },
    {
      "id": "S047",
      "uses": [
        "G001",
        "S046"
      ],
      "before": "positive divisor candidate23:a=23;b=30;quotient=1;remainder=7",
      "after": "divides=否;witness/status=不存在",
      "relation": "implies",
      "rule_id": "positive_divisor_remainder_criterion",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正a下，餘數0時商就是自然乘數見證；非零餘數由商餘唯一性排除整除，不把截斷商當見證。",
        "en": "For positive a,a zero remainder makes the quotient a natural multiplier witness.A nonzero remainder rules out divisibility by uniqueness;the truncated quotient is not a witness."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "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": "positive divisor candidate24:divide natural 30 by positive 24",
      "after": "30=1·24+6",
      "relation": "implies",
      "rule_id": "natural_euclidean_division_witness",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數給唯一商餘，餘數滿足0≤r<divisor；這個完整等式同時核商與餘數，不作除以0。",
        "en": "A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "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",
          "24",
          "1"
        ],
        "output": "6",
        "label": "positive divisor candidate24"
      }
    },
    {
      "id": "S049",
      "uses": [
        "G001",
        "S048"
      ],
      "before": "positive divisor candidate24:a=24;b=30;quotient=1;remainder=6",
      "after": "divides=否;witness/status=不存在",
      "relation": "implies",
      "rule_id": "positive_divisor_remainder_criterion",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正a下，餘數0時商就是自然乘數見證；非零餘數由商餘唯一性排除整除，不把截斷商當見證。",
        "en": "For positive a,a zero remainder makes the quotient a natural multiplier witness.A nonzero remainder rules out divisibility by uniqueness;the truncated quotient is not a witness."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "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": "positive divisor candidate25:divide natural 30 by positive 25",
      "after": "30=1·25+5",
      "relation": "implies",
      "rule_id": "natural_euclidean_division_witness",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數給唯一商餘，餘數滿足0≤r<divisor；這個完整等式同時核商與餘數，不作除以0。",
        "en": "A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "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",
          "25",
          "1"
        ],
        "output": "5",
        "label": "positive divisor candidate25"
      }
    },
    {
      "id": "S051",
      "uses": [
        "G001",
        "S050"
      ],
      "before": "positive divisor candidate25:a=25;b=30;quotient=1;remainder=5",
      "after": "divides=否;witness/status=不存在",
      "relation": "implies",
      "rule_id": "positive_divisor_remainder_criterion",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正a下，餘數0時商就是自然乘數見證；非零餘數由商餘唯一性排除整除，不把截斷商當見證。",
        "en": "For positive a,a zero remainder makes the quotient a natural multiplier witness.A nonzero remainder rules out divisibility by uniqueness;the truncated quotient is not a witness."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "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": "positive divisor candidate26:divide natural 30 by positive 26",
      "after": "30=1·26+4",
      "relation": "implies",
      "rule_id": "natural_euclidean_division_witness",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數給唯一商餘，餘數滿足0≤r<divisor；這個完整等式同時核商與餘數，不作除以0。",
        "en": "A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "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",
          "26",
          "1"
        ],
        "output": "4",
        "label": "positive divisor candidate26"
      }
    },
    {
      "id": "S053",
      "uses": [
        "G001",
        "S052"
      ],
      "before": "positive divisor candidate26:a=26;b=30;quotient=1;remainder=4",
      "after": "divides=否;witness/status=不存在",
      "relation": "implies",
      "rule_id": "positive_divisor_remainder_criterion",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正a下，餘數0時商就是自然乘數見證；非零餘數由商餘唯一性排除整除，不把截斷商當見證。",
        "en": "For positive a,a zero remainder makes the quotient a natural multiplier witness.A nonzero remainder rules out divisibility by uniqueness;the truncated quotient is not a witness."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "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": "positive divisor candidate27:divide natural 30 by positive 27",
      "after": "30=1·27+3",
      "relation": "implies",
      "rule_id": "natural_euclidean_division_witness",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數給唯一商餘，餘數滿足0≤r<divisor；這個完整等式同時核商與餘數，不作除以0。",
        "en": "A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "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",
          "27",
          "1"
        ],
        "output": "3",
        "label": "positive divisor candidate27"
      }
    },
    {
      "id": "S055",
      "uses": [
        "G001",
        "S054"
      ],
      "before": "positive divisor candidate27:a=27;b=30;quotient=1;remainder=3",
      "after": "divides=否;witness/status=不存在",
      "relation": "implies",
      "rule_id": "positive_divisor_remainder_criterion",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正a下，餘數0時商就是自然乘數見證；非零餘數由商餘唯一性排除整除，不把截斷商當見證。",
        "en": "For positive a,a zero remainder makes the quotient a natural multiplier witness.A nonzero remainder rules out divisibility by uniqueness;the truncated quotient is not a witness."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S056",
      "uses": [
        "G001",
        "S055"
      ],
      "before": "positive divisor candidate28:divide natural 30 by positive 28",
      "after": "30=1·28+2",
      "relation": "implies",
      "rule_id": "natural_euclidean_division_witness",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數給唯一商餘，餘數滿足0≤r<divisor；這個完整等式同時核商與餘數，不作除以0。",
        "en": "A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "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",
          "28",
          "1"
        ],
        "output": "2",
        "label": "positive divisor candidate28"
      }
    },
    {
      "id": "S057",
      "uses": [
        "G001",
        "S056"
      ],
      "before": "positive divisor candidate28:a=28;b=30;quotient=1;remainder=2",
      "after": "divides=否;witness/status=不存在",
      "relation": "implies",
      "rule_id": "positive_divisor_remainder_criterion",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正a下，餘數0時商就是自然乘數見證；非零餘數由商餘唯一性排除整除，不把截斷商當見證。",
        "en": "For positive a,a zero remainder makes the quotient a natural multiplier witness.A nonzero remainder rules out divisibility by uniqueness;the truncated quotient is not a witness."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "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": "positive divisor candidate29:divide natural 30 by positive 29",
      "after": "30=1·29+1",
      "relation": "implies",
      "rule_id": "natural_euclidean_division_witness",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數給唯一商餘，餘數滿足0≤r<divisor；這個完整等式同時核商與餘數，不作除以0。",
        "en": "A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "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",
          "29",
          "1"
        ],
        "output": "1",
        "label": "positive divisor candidate29"
      }
    },
    {
      "id": "S059",
      "uses": [
        "G001",
        "S058"
      ],
      "before": "positive divisor candidate29:a=29;b=30;quotient=1;remainder=1",
      "after": "divides=否;witness/status=不存在",
      "relation": "implies",
      "rule_id": "positive_divisor_remainder_criterion",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正a下，餘數0時商就是自然乘數見證；非零餘數由商餘唯一性排除整除，不把截斷商當見證。",
        "en": "For positive a,a zero remainder makes the quotient a natural multiplier witness.A nonzero remainder rules out divisibility by uniqueness;the truncated quotient is not a witness."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "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": "positive divisor candidate30:divide natural 30 by positive 30",
      "after": "30=1·30+0",
      "relation": "implies",
      "rule_id": "natural_euclidean_division_witness",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正除數給唯一商餘，餘數滿足0≤r<divisor；這個完整等式同時核商與餘數，不作除以0。",
        "en": "A positive divisor gives a unique quotient/remainder with0≤r<divisor.The full equality audits both values without division by zero."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "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",
          "30",
          "1"
        ],
        "output": "0",
        "label": "positive divisor candidate30"
      }
    },
    {
      "id": "S061",
      "uses": [
        "G001",
        "S060"
      ],
      "before": "positive divisor candidate30:a=30;b=30;quotient=1;remainder=0",
      "after": "divides=是;witness/status=1",
      "relation": "implies",
      "rule_id": "positive_divisor_remainder_criterion",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正a下，餘數0時商就是自然乘數見證；非零餘數由商餘唯一性排除整除，不把截斷商當見證。",
        "en": "For positive a,a zero remainder makes the quotient a natural multiplier witness.A nonzero remainder rules out divisibility by uniqueness;the truncated quotient is not a witness."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S062",
      "uses": [
        "G001",
        "S061"
      ],
      "before": "positive b=30;complete candidate interval=[1,30]",
      "after": "positive divisors=[1, 2, 3, 5, 6, 10, 15, 30];count=8",
      "relation": "implies",
      "rule_id": "complete_positive_divisor_bound",
      "conditions": [],
      "explanation": {
        "zh-Hant": "若正d整除正b，b=d*k且k≥1，故d≤b；因此1到b是完整候選域，不是只抽樣找到部分因數。",
        "en": "If positive d divides positive b,then b=d*k with k≥1,so d≤b.Thus1 through b is the complete candidate domain,not a sample of some factors."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "result": [
      30,
      [
        1,
        2,
        3,
        5,
        6,
        10,
        15,
        30
      ],
      8
    ]
  },
  "conclusion": {
    "canonical_value": "(30, [1, 2, 3, 5, 6, 10, 15, 30], 8)",
    "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-r02-natural-divisibility-foundations",
      "batch_path": "public/data/questions/batches/batch-20261013-advance-20261018-r02-natural-divisibility-foundations/questions.jsonl",
      "batch_sha256": "85abf07ffd0c46a8235728cd1ab694010d535a2683ea0040042f9de7a1360357",
      "original_line": 46,
      "original_line_sha256": "a3ac402cb7b871098f0f86f473c043801c2c004dcceaae433748c0697646af58",
      "question_compact_sha256": "7d61c0446c07279e818853d34cef67aaeb1ece20f6db90830085da19999e4b5c",
      "english_row_compact_sha256": "038010d3ce2422964f6e6862455bb53881048c736912fb4ca9f12e1bc9179801",
      "english_manifest_path": "public/data/i18n/editions/batch-20261013-advance-20261018-r02-natural-divisibility-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261013-advance-20261018-r02-natural-divisibility-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-natural-divisibility": {
        "sha256": "2e5d00faa3ae3031b2cb2a650b12db0ab8e8aa1b59a7ea6b2350f49a792758d7",
        "english_sha256": "acae542d73cc84e0fd3fd3443b35f99bca7758dce17f8b8ab65ee9957193e395",
        "original_path": "public/data/mko/natural-divisibility.json",
        "english_path": "public/data/i18n/editions/mko-natural-divisibility.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "8e8954cf690c44da1ec1efa08994a485c73ca8706b7865d854e3f2cdfd4ca89f",
    "original_candidate_sha256": "977fb89f380ed8b72596d0c0cc1a97c273bf5d4193be1028460e7e68a58619cb",
    "original_candidate_line_sha256": "977fb89f380ed8b72596d0c0cc1a97c273bf5d4193be1028460e7e68a58619cb"
  },
  "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-20261026-005-p17",
    "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": "00fcddb10a9fb0f56697308976a52e68e2537472bd8e68ecbe96d3c4553f4ad2",
      "question_first_witness_sha256": "198e04380b2d32aef2375b07ca0df1f8afa454d1184993173fad8a3c9187a3d1"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 62,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "a8fab51d109be57c1085894c3c9eeff8dda2ec4c3603d6ff10a2b116917a89b4",
      "arithmetic_steps": 30
    },
    "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."
  }
}