{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261013-advance-20261018-r02-natural-divisibility-0071",
  "pilot_number": "R0016-08796",
  "number": 8796,
  "family_id": "qf-ext18-divisibility-common-multiples",
  "title": {
    "zh-Hant": "第 8796 題：自然數整除與零邊界",
    "en": "Question 8796: All Common Multiples in an Inclusive Natural Window"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261013-advance-20261018-r02-natural-divisibility-0071",
    "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-common-multiples；資料={\"a\":6,\"b\":4,\"bound\":50}。請依 [a,b,bound,common_multiples] 的欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(6, 4, 50, [0, 12, 24, 36, 48])",
      "explanation_zh": "擴展基礎核對=(6, 4, 50, [0, 12, 24, 36, 48])；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-common-multiples",
      "seed": 202610182071,
      "parameters": {
        "a": 6,
        "b": 4,
        "bound": 50,
        "domain": "Nat_divisibility",
        "operation": "divisibility-common-multiples",
        "semantic_template": "qf-ext18-divisibility-common-multiples/{\"a\":6,\"b\":4,\"bound\":50}"
      }
    },
    "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-0071",
    "batch_id": "batch-20261013-advance-20261018-r02-natural-divisibility-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "0d7f06a016245e07167922f0b686d8bdbc67e31964e21bb8566c8c15e7025a6b",
      "family_id": "qf-ext18-divisibility-common-multiples",
      "seed": 202610182071,
      "semantic_template": "qf-ext18-divisibility-common-multiples/{\"a\":6,\"b\":4,\"bound\":50}"
    },
    "fields": {
      "title": "All Common Multiples in an Inclusive Natural Window",
      "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. a=6, b=4, bound=50. Give (a, b, bound, all common multiples in [0,bound] in increasing order).",
      "answer": {
        "kind": "expression",
        "value": "(6, 4, 50, [0, 12, 24, 36, 48])",
        "explanation": "Inspect the complete inclusive window and apply both divisibility guards; zero is retained when appropriate. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, 4, 50, [0, 12, 24, 36, 48])."
      },
      "skills": [
        "All Common Multiples in an Inclusive Natural Window"
      ],
      "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-common-multiples；資料={\"a\":6,\"b\":4,\"bound\":50}。請依 [a,b,bound,common_multiples] 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. a=6, b=4, bound=50. Give (a, b, bound, all common multiples in [0,bound] in increasing order)."
      }
    }
  ],
  "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": "window0 first divisor:divide natural 0 by positive 6",
      "after": "0=0·6+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": [
          "0",
          "6",
          "0"
        ],
        "output": "0",
        "label": "window0 first divisor"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "window0 first divisor:a=6;b=0;quotient=0;remainder=0",
      "after": "divides=是;witness/status=0",
      "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": "window0 second divisor:divide natural 0 by positive 4",
      "after": "0=0·4+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": [
          "0",
          "4",
          "0"
        ],
        "output": "0",
        "label": "window0 second divisor"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "window0 second divisor:a=4;b=0;quotient=0;remainder=0",
      "after": "divides=是;witness/status=0",
      "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": "candidate=0;a divides=是;b divides=是",
      "after": "common multiple=是",
      "relation": "implies",
      "rule_id": "both_divisibility_window_guards",
      "conditions": [],
      "explanation": {
        "zh-Hant": "同一候選必須同時被a與b整除；完整包含0和bound端點，遇零除數也依原整除定義處理。",
        "en": "The same candidate must be divisible by both a and b.Include0 and the bound endpoint,and handle zero divisors using the original divisibility definition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "window1 first divisor:divide natural 1 by positive 6",
      "after": "1=0·6+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": [
          "1",
          "6",
          "0"
        ],
        "output": "1",
        "label": "window1 first divisor"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "window1 first divisor:a=6;b=1;quotient=0;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": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "window1 second divisor:divide natural 1 by positive 4",
      "after": "1=0·4+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": [
          "1",
          "4",
          "0"
        ],
        "output": "1",
        "label": "window1 second divisor"
      }
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "window1 second divisor:a=4;b=1;quotient=0;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": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "candidate=1;a divides=否;b divides=否",
      "after": "common multiple=否",
      "relation": "implies",
      "rule_id": "both_divisibility_window_guards",
      "conditions": [],
      "explanation": {
        "zh-Hant": "同一候選必須同時被a與b整除；完整包含0和bound端點，遇零除數也依原整除定義處理。",
        "en": "The same candidate must be divisible by both a and b.Include0 and the bound endpoint,and handle zero divisors using the original divisibility definition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "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": "window2 first divisor:divide natural 2 by positive 6",
      "after": "2=0·6+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": [
          "2",
          "6",
          "0"
        ],
        "output": "2",
        "label": "window2 first divisor"
      }
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "window2 first divisor:a=6;b=2;quotient=0;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": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "window2 second divisor:divide natural 2 by positive 4",
      "after": "2=0·4+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": [
          "2",
          "4",
          "0"
        ],
        "output": "2",
        "label": "window2 second divisor"
      }
    },
    {
      "id": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "window2 second divisor:a=4;b=2;quotient=0;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": "candidate=2;a divides=否;b divides=否",
      "after": "common multiple=否",
      "relation": "implies",
      "rule_id": "both_divisibility_window_guards",
      "conditions": [],
      "explanation": {
        "zh-Hant": "同一候選必須同時被a與b整除；完整包含0和bound端點，遇零除數也依原整除定義處理。",
        "en": "The same candidate must be divisible by both a and b.Include0 and the bound endpoint,and handle zero divisors using the original divisibility definition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "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": "window3 first divisor:divide natural 3 by positive 6",
      "after": "3=0·6+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": [
          "3",
          "6",
          "0"
        ],
        "output": "3",
        "label": "window3 first divisor"
      }
    },
    {
      "id": "S018",
      "uses": [
        "G001",
        "S017"
      ],
      "before": "window3 first divisor:a=6;b=3;quotient=0;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": "S019",
      "uses": [
        "G001",
        "S018"
      ],
      "before": "window3 second divisor:divide natural 3 by positive 4",
      "after": "3=0·4+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": [
          "3",
          "4",
          "0"
        ],
        "output": "3",
        "label": "window3 second divisor"
      }
    },
    {
      "id": "S020",
      "uses": [
        "G001",
        "S019"
      ],
      "before": "window3 second divisor:a=4;b=3;quotient=0;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": "S021",
      "uses": [
        "G001",
        "S020"
      ],
      "before": "candidate=3;a divides=否;b divides=否",
      "after": "common multiple=否",
      "relation": "implies",
      "rule_id": "both_divisibility_window_guards",
      "conditions": [],
      "explanation": {
        "zh-Hant": "同一候選必須同時被a與b整除；完整包含0和bound端點，遇零除數也依原整除定義處理。",
        "en": "The same candidate must be divisible by both a and b.Include0 and the bound endpoint,and handle zero divisors using the original divisibility definition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "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": "window4 first divisor:divide natural 4 by positive 6",
      "after": "4=0·6+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": [
          "4",
          "6",
          "0"
        ],
        "output": "4",
        "label": "window4 first divisor"
      }
    },
    {
      "id": "S023",
      "uses": [
        "G001",
        "S022"
      ],
      "before": "window4 first divisor:a=6;b=4;quotient=0;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": "S024",
      "uses": [
        "G001",
        "S023"
      ],
      "before": "window4 second divisor:divide natural 4 by positive 4",
      "after": "4=1·4+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": [
          "4",
          "4",
          "1"
        ],
        "output": "0",
        "label": "window4 second divisor"
      }
    },
    {
      "id": "S025",
      "uses": [
        "G001",
        "S024"
      ],
      "before": "window4 second divisor:a=4;b=4;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": "S026",
      "uses": [
        "G001",
        "S025"
      ],
      "before": "candidate=4;a divides=否;b divides=是",
      "after": "common multiple=否",
      "relation": "implies",
      "rule_id": "both_divisibility_window_guards",
      "conditions": [],
      "explanation": {
        "zh-Hant": "同一候選必須同時被a與b整除；完整包含0和bound端點，遇零除數也依原整除定義處理。",
        "en": "The same candidate must be divisible by both a and b.Include0 and the bound endpoint,and handle zero divisors using the original divisibility definition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "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": "window5 first divisor:divide natural 5 by positive 6",
      "after": "5=0·6+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": [
          "5",
          "6",
          "0"
        ],
        "output": "5",
        "label": "window5 first divisor"
      }
    },
    {
      "id": "S028",
      "uses": [
        "G001",
        "S027"
      ],
      "before": "window5 first divisor:a=6;b=5;quotient=0;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": "S029",
      "uses": [
        "G001",
        "S028"
      ],
      "before": "window5 second divisor:divide natural 5 by positive 4",
      "after": "5=1·4+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": [
          "5",
          "4",
          "1"
        ],
        "output": "1",
        "label": "window5 second divisor"
      }
    },
    {
      "id": "S030",
      "uses": [
        "G001",
        "S029"
      ],
      "before": "window5 second divisor:a=4;b=5;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": "S031",
      "uses": [
        "G001",
        "S030"
      ],
      "before": "candidate=5;a divides=否;b divides=否",
      "after": "common multiple=否",
      "relation": "implies",
      "rule_id": "both_divisibility_window_guards",
      "conditions": [],
      "explanation": {
        "zh-Hant": "同一候選必須同時被a與b整除；完整包含0和bound端點，遇零除數也依原整除定義處理。",
        "en": "The same candidate must be divisible by both a and b.Include0 and the bound endpoint,and handle zero divisors using the original divisibility definition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "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": "window6 first divisor:divide natural 6 by positive 6",
      "after": "6=1·6+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": [
          "6",
          "6",
          "1"
        ],
        "output": "0",
        "label": "window6 first divisor"
      }
    },
    {
      "id": "S033",
      "uses": [
        "G001",
        "S032"
      ],
      "before": "window6 first divisor:a=6;b=6;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": "S034",
      "uses": [
        "G001",
        "S033"
      ],
      "before": "window6 second divisor:divide natural 6 by positive 4",
      "after": "6=1·4+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": [
          "6",
          "4",
          "1"
        ],
        "output": "2",
        "label": "window6 second divisor"
      }
    },
    {
      "id": "S035",
      "uses": [
        "G001",
        "S034"
      ],
      "before": "window6 second divisor:a=4;b=6;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": "S036",
      "uses": [
        "G001",
        "S035"
      ],
      "before": "candidate=6;a divides=是;b divides=否",
      "after": "common multiple=否",
      "relation": "implies",
      "rule_id": "both_divisibility_window_guards",
      "conditions": [],
      "explanation": {
        "zh-Hant": "同一候選必須同時被a與b整除；完整包含0和bound端點，遇零除數也依原整除定義處理。",
        "en": "The same candidate must be divisible by both a and b.Include0 and the bound endpoint,and handle zero divisors using the original divisibility definition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S037",
      "uses": [
        "G001",
        "S036"
      ],
      "before": "window7 first divisor:divide natural 7 by positive 6",
      "after": "7=1·6+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": [
          "7",
          "6",
          "1"
        ],
        "output": "1",
        "label": "window7 first divisor"
      }
    },
    {
      "id": "S038",
      "uses": [
        "G001",
        "S037"
      ],
      "before": "window7 first divisor:a=6;b=7;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": "S039",
      "uses": [
        "G001",
        "S038"
      ],
      "before": "window7 second divisor:divide natural 7 by positive 4",
      "after": "7=1·4+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": [
          "7",
          "4",
          "1"
        ],
        "output": "3",
        "label": "window7 second divisor"
      }
    },
    {
      "id": "S040",
      "uses": [
        "G001",
        "S039"
      ],
      "before": "window7 second divisor:a=4;b=7;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": "S041",
      "uses": [
        "G001",
        "S040"
      ],
      "before": "candidate=7;a divides=否;b divides=否",
      "after": "common multiple=否",
      "relation": "implies",
      "rule_id": "both_divisibility_window_guards",
      "conditions": [],
      "explanation": {
        "zh-Hant": "同一候選必須同時被a與b整除；完整包含0和bound端點，遇零除數也依原整除定義處理。",
        "en": "The same candidate must be divisible by both a and b.Include0 and the bound endpoint,and handle zero divisors using the original divisibility definition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "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": "window8 first divisor:divide natural 8 by positive 6",
      "after": "8=1·6+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": [
          "8",
          "6",
          "1"
        ],
        "output": "2",
        "label": "window8 first divisor"
      }
    },
    {
      "id": "S043",
      "uses": [
        "G001",
        "S042"
      ],
      "before": "window8 first divisor:a=6;b=8;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": "S044",
      "uses": [
        "G001",
        "S043"
      ],
      "before": "window8 second divisor:divide natural 8 by positive 4",
      "after": "8=2·4+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": [
          "8",
          "4",
          "2"
        ],
        "output": "0",
        "label": "window8 second divisor"
      }
    },
    {
      "id": "S045",
      "uses": [
        "G001",
        "S044"
      ],
      "before": "window8 second divisor:a=4;b=8;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": "S046",
      "uses": [
        "G001",
        "S045"
      ],
      "before": "candidate=8;a divides=否;b divides=是",
      "after": "common multiple=否",
      "relation": "implies",
      "rule_id": "both_divisibility_window_guards",
      "conditions": [],
      "explanation": {
        "zh-Hant": "同一候選必須同時被a與b整除；完整包含0和bound端點，遇零除數也依原整除定義處理。",
        "en": "The same candidate must be divisible by both a and b.Include0 and the bound endpoint,and handle zero divisors using the original divisibility definition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "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": "window9 first divisor:divide natural 9 by positive 6",
      "after": "9=1·6+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": [
          "9",
          "6",
          "1"
        ],
        "output": "3",
        "label": "window9 first divisor"
      }
    },
    {
      "id": "S048",
      "uses": [
        "G001",
        "S047"
      ],
      "before": "window9 first divisor:a=6;b=9;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": "S049",
      "uses": [
        "G001",
        "S048"
      ],
      "before": "window9 second divisor:divide natural 9 by positive 4",
      "after": "9=2·4+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": [
          "9",
          "4",
          "2"
        ],
        "output": "1",
        "label": "window9 second divisor"
      }
    },
    {
      "id": "S050",
      "uses": [
        "G001",
        "S049"
      ],
      "before": "window9 second divisor:a=4;b=9;quotient=2;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": "S051",
      "uses": [
        "G001",
        "S050"
      ],
      "before": "candidate=9;a divides=否;b divides=否",
      "after": "common multiple=否",
      "relation": "implies",
      "rule_id": "both_divisibility_window_guards",
      "conditions": [],
      "explanation": {
        "zh-Hant": "同一候選必須同時被a與b整除；完整包含0和bound端點，遇零除數也依原整除定義處理。",
        "en": "The same candidate must be divisible by both a and b.Include0 and the bound endpoint,and handle zero divisors using the original divisibility definition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "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": "window10 first divisor:divide natural 10 by positive 6",
      "after": "10=1·6+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": [
          "10",
          "6",
          "1"
        ],
        "output": "4",
        "label": "window10 first divisor"
      }
    },
    {
      "id": "S053",
      "uses": [
        "G001",
        "S052"
      ],
      "before": "window10 first divisor:a=6;b=10;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": "window10 second divisor:divide natural 10 by positive 4",
      "after": "10=2·4+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": [
          "10",
          "4",
          "2"
        ],
        "output": "2",
        "label": "window10 second divisor"
      }
    },
    {
      "id": "S055",
      "uses": [
        "G001",
        "S054"
      ],
      "before": "window10 second divisor:a=4;b=10;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": "S056",
      "uses": [
        "G001",
        "S055"
      ],
      "before": "candidate=10;a divides=否;b divides=否",
      "after": "common multiple=否",
      "relation": "implies",
      "rule_id": "both_divisibility_window_guards",
      "conditions": [],
      "explanation": {
        "zh-Hant": "同一候選必須同時被a與b整除；完整包含0和bound端點，遇零除數也依原整除定義處理。",
        "en": "The same candidate must be divisible by both a and b.Include0 and the bound endpoint,and handle zero divisors using the original divisibility definition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "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": "window11 first divisor:divide natural 11 by positive 6",
      "after": "11=1·6+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": [
          "11",
          "6",
          "1"
        ],
        "output": "5",
        "label": "window11 first divisor"
      }
    },
    {
      "id": "S058",
      "uses": [
        "G001",
        "S057"
      ],
      "before": "window11 first divisor:a=6;b=11;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": "S059",
      "uses": [
        "G001",
        "S058"
      ],
      "before": "window11 second divisor:divide natural 11 by positive 4",
      "after": "11=2·4+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": [
          "11",
          "4",
          "2"
        ],
        "output": "3",
        "label": "window11 second divisor"
      }
    },
    {
      "id": "S060",
      "uses": [
        "G001",
        "S059"
      ],
      "before": "window11 second divisor:a=4;b=11;quotient=2;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": "S061",
      "uses": [
        "G001",
        "S060"
      ],
      "before": "candidate=11;a divides=否;b divides=否",
      "after": "common multiple=否",
      "relation": "implies",
      "rule_id": "both_divisibility_window_guards",
      "conditions": [],
      "explanation": {
        "zh-Hant": "同一候選必須同時被a與b整除；完整包含0和bound端點，遇零除數也依原整除定義處理。",
        "en": "The same candidate must be divisible by both a and b.Include0 and the bound endpoint,and handle zero divisors using the original divisibility definition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "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": "window12 first divisor:divide natural 12 by positive 6",
      "after": "12=2·6+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": [
          "12",
          "6",
          "2"
        ],
        "output": "0",
        "label": "window12 first divisor"
      }
    },
    {
      "id": "S063",
      "uses": [
        "G001",
        "S062"
      ],
      "before": "window12 first divisor:a=6;b=12;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": "S064",
      "uses": [
        "G001",
        "S063"
      ],
      "before": "window12 second divisor:divide natural 12 by positive 4",
      "after": "12=3·4+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": [
          "12",
          "4",
          "3"
        ],
        "output": "0",
        "label": "window12 second divisor"
      }
    },
    {
      "id": "S065",
      "uses": [
        "G001",
        "S064"
      ],
      "before": "window12 second divisor:a=4;b=12;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": "S066",
      "uses": [
        "G001",
        "S065"
      ],
      "before": "candidate=12;a divides=是;b divides=是",
      "after": "common multiple=是",
      "relation": "implies",
      "rule_id": "both_divisibility_window_guards",
      "conditions": [],
      "explanation": {
        "zh-Hant": "同一候選必須同時被a與b整除；完整包含0和bound端點，遇零除數也依原整除定義處理。",
        "en": "The same candidate must be divisible by both a and b.Include0 and the bound endpoint,and handle zero divisors using the original divisibility definition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "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": "window13 first divisor:divide natural 13 by positive 6",
      "after": "13=2·6+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": [
          "13",
          "6",
          "2"
        ],
        "output": "1",
        "label": "window13 first divisor"
      }
    },
    {
      "id": "S068",
      "uses": [
        "G001",
        "S067"
      ],
      "before": "window13 first divisor:a=6;b=13;quotient=2;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": "S069",
      "uses": [
        "G001",
        "S068"
      ],
      "before": "window13 second divisor:divide natural 13 by positive 4",
      "after": "13=3·4+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": [
          "13",
          "4",
          "3"
        ],
        "output": "1",
        "label": "window13 second divisor"
      }
    },
    {
      "id": "S070",
      "uses": [
        "G001",
        "S069"
      ],
      "before": "window13 second divisor:a=4;b=13;quotient=3;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": "S071",
      "uses": [
        "G001",
        "S070"
      ],
      "before": "candidate=13;a divides=否;b divides=否",
      "after": "common multiple=否",
      "relation": "implies",
      "rule_id": "both_divisibility_window_guards",
      "conditions": [],
      "explanation": {
        "zh-Hant": "同一候選必須同時被a與b整除；完整包含0和bound端點，遇零除數也依原整除定義處理。",
        "en": "The same candidate must be divisible by both a and b.Include0 and the bound endpoint,and handle zero divisors using the original divisibility definition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S072",
      "uses": [
        "G001",
        "S071"
      ],
      "before": "window14 first divisor:divide natural 14 by positive 6",
      "after": "14=2·6+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": [
          "14",
          "6",
          "2"
        ],
        "output": "2",
        "label": "window14 first divisor"
      }
    },
    {
      "id": "S073",
      "uses": [
        "G001",
        "S072"
      ],
      "before": "window14 first divisor:a=6;b=14;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": "S074",
      "uses": [
        "G001",
        "S073"
      ],
      "before": "window14 second divisor:divide natural 14 by positive 4",
      "after": "14=3·4+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": [
          "14",
          "4",
          "3"
        ],
        "output": "2",
        "label": "window14 second divisor"
      }
    },
    {
      "id": "S075",
      "uses": [
        "G001",
        "S074"
      ],
      "before": "window14 second divisor:a=4;b=14;quotient=3;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": "S076",
      "uses": [
        "G001",
        "S075"
      ],
      "before": "candidate=14;a divides=否;b divides=否",
      "after": "common multiple=否",
      "relation": "implies",
      "rule_id": "both_divisibility_window_guards",
      "conditions": [],
      "explanation": {
        "zh-Hant": "同一候選必須同時被a與b整除；完整包含0和bound端點，遇零除數也依原整除定義處理。",
        "en": "The same candidate must be divisible by both a and b.Include0 and the bound endpoint,and handle zero divisors using the original divisibility definition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S077",
      "uses": [
        "G001",
        "S076"
      ],
      "before": "window15 first divisor:divide natural 15 by positive 6",
      "after": "15=2·6+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": [
          "15",
          "6",
          "2"
        ],
        "output": "3",
        "label": "window15 first divisor"
      }
    },
    {
      "id": "S078",
      "uses": [
        "G001",
        "S077"
      ],
      "before": "window15 first divisor:a=6;b=15;quotient=2;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": "S079",
      "uses": [
        "G001",
        "S078"
      ],
      "before": "window15 second divisor:divide natural 15 by positive 4",
      "after": "15=3·4+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": [
          "15",
          "4",
          "3"
        ],
        "output": "3",
        "label": "window15 second divisor"
      }
    },
    {
      "id": "S080",
      "uses": [
        "G001",
        "S079"
      ],
      "before": "window15 second divisor:a=4;b=15;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": "S081",
      "uses": [
        "G001",
        "S080"
      ],
      "before": "candidate=15;a divides=否;b divides=否",
      "after": "common multiple=否",
      "relation": "implies",
      "rule_id": "both_divisibility_window_guards",
      "conditions": [],
      "explanation": {
        "zh-Hant": "同一候選必須同時被a與b整除；完整包含0和bound端點，遇零除數也依原整除定義處理。",
        "en": "The same candidate must be divisible by both a and b.Include0 and the bound endpoint,and handle zero divisors using the original divisibility definition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "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": "window16 first divisor:divide natural 16 by positive 6",
      "after": "16=2·6+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": [
          "16",
          "6",
          "2"
        ],
        "output": "4",
        "label": "window16 first divisor"
      }
    },
    {
      "id": "S083",
      "uses": [
        "G001",
        "S082"
      ],
      "before": "window16 first divisor:a=6;b=16;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": "S084",
      "uses": [
        "G001",
        "S083"
      ],
      "before": "window16 second divisor:divide natural 16 by positive 4",
      "after": "16=4·4+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": [
          "16",
          "4",
          "4"
        ],
        "output": "0",
        "label": "window16 second divisor"
      }
    },
    {
      "id": "S085",
      "uses": [
        "G001",
        "S084"
      ],
      "before": "window16 second divisor:a=4;b=16;quotient=4;remainder=0",
      "after": "divides=是;witness/status=4",
      "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": "S086",
      "uses": [
        "G001",
        "S085"
      ],
      "before": "candidate=16;a divides=否;b divides=是",
      "after": "common multiple=否",
      "relation": "implies",
      "rule_id": "both_divisibility_window_guards",
      "conditions": [],
      "explanation": {
        "zh-Hant": "同一候選必須同時被a與b整除；完整包含0和bound端點，遇零除數也依原整除定義處理。",
        "en": "The same candidate must be divisible by both a and b.Include0 and the bound endpoint,and handle zero divisors using the original divisibility definition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S087",
      "uses": [
        "G001",
        "S086"
      ],
      "before": "window17 first divisor:divide natural 17 by positive 6",
      "after": "17=2·6+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": [
          "17",
          "6",
          "2"
        ],
        "output": "5",
        "label": "window17 first divisor"
      }
    },
    {
      "id": "S088",
      "uses": [
        "G001",
        "S087"
      ],
      "before": "window17 first divisor:a=6;b=17;quotient=2;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": "S089",
      "uses": [
        "G001",
        "S088"
      ],
      "before": "window17 second divisor:divide natural 17 by positive 4",
      "after": "17=4·4+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": [
          "17",
          "4",
          "4"
        ],
        "output": "1",
        "label": "window17 second divisor"
      }
    },
    {
      "id": "S090",
      "uses": [
        "G001",
        "S089"
      ],
      "before": "window17 second divisor:a=4;b=17;quotient=4;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": "S091",
      "uses": [
        "G001",
        "S090"
      ],
      "before": "candidate=17;a divides=否;b divides=否",
      "after": "common multiple=否",
      "relation": "implies",
      "rule_id": "both_divisibility_window_guards",
      "conditions": [],
      "explanation": {
        "zh-Hant": "同一候選必須同時被a與b整除；完整包含0和bound端點，遇零除數也依原整除定義處理。",
        "en": "The same candidate must be divisible by both a and b.Include0 and the bound endpoint,and handle zero divisors using the original divisibility definition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S092",
      "uses": [
        "G001",
        "S091"
      ],
      "before": "window18 first divisor:divide natural 18 by positive 6",
      "after": "18=3·6+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": [
          "18",
          "6",
          "3"
        ],
        "output": "0",
        "label": "window18 first divisor"
      }
    },
    {
      "id": "S093",
      "uses": [
        "G001",
        "S092"
      ],
      "before": "window18 first divisor:a=6;b=18;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": "S094",
      "uses": [
        "G001",
        "S093"
      ],
      "before": "window18 second divisor:divide natural 18 by positive 4",
      "after": "18=4·4+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": [
          "18",
          "4",
          "4"
        ],
        "output": "2",
        "label": "window18 second divisor"
      }
    },
    {
      "id": "S095",
      "uses": [
        "G001",
        "S094"
      ],
      "before": "window18 second divisor:a=4;b=18;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": "S096",
      "uses": [
        "G001",
        "S095"
      ],
      "before": "candidate=18;a divides=是;b divides=否",
      "after": "common multiple=否",
      "relation": "implies",
      "rule_id": "both_divisibility_window_guards",
      "conditions": [],
      "explanation": {
        "zh-Hant": "同一候選必須同時被a與b整除；完整包含0和bound端點，遇零除數也依原整除定義處理。",
        "en": "The same candidate must be divisible by both a and b.Include0 and the bound endpoint,and handle zero divisors using the original divisibility definition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S097",
      "uses": [
        "G001",
        "S096"
      ],
      "before": "window19 first divisor:divide natural 19 by positive 6",
      "after": "19=3·6+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": [
          "19",
          "6",
          "3"
        ],
        "output": "1",
        "label": "window19 first divisor"
      }
    },
    {
      "id": "S098",
      "uses": [
        "G001",
        "S097"
      ],
      "before": "window19 first divisor:a=6;b=19;quotient=3;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": "S099",
      "uses": [
        "G001",
        "S098"
      ],
      "before": "window19 second divisor:divide natural 19 by positive 4",
      "after": "19=4·4+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": [
          "19",
          "4",
          "4"
        ],
        "output": "3",
        "label": "window19 second divisor"
      }
    },
    {
      "id": "S100",
      "uses": [
        "G001",
        "S099"
      ],
      "before": "window19 second divisor:a=4;b=19;quotient=4;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": "S101",
      "uses": [
        "G001",
        "S100"
      ],
      "before": "candidate=19;a divides=否;b divides=否",
      "after": "common multiple=否",
      "relation": "implies",
      "rule_id": "both_divisibility_window_guards",
      "conditions": [],
      "explanation": {
        "zh-Hant": "同一候選必須同時被a與b整除；完整包含0和bound端點，遇零除數也依原整除定義處理。",
        "en": "The same candidate must be divisible by both a and b.Include0 and the bound endpoint,and handle zero divisors using the original divisibility definition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S102",
      "uses": [
        "G001",
        "S101"
      ],
      "before": "window20 first divisor:divide natural 20 by positive 6",
      "after": "20=3·6+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": [
          "20",
          "6",
          "3"
        ],
        "output": "2",
        "label": "window20 first divisor"
      }
    },
    {
      "id": "S103",
      "uses": [
        "G001",
        "S102"
      ],
      "before": "window20 first divisor:a=6;b=20;quotient=3;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": "S104",
      "uses": [
        "G001",
        "S103"
      ],
      "before": "window20 second divisor:divide natural 20 by positive 4",
      "after": "20=5·4+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": [
          "20",
          "4",
          "5"
        ],
        "output": "0",
        "label": "window20 second divisor"
      }
    },
    {
      "id": "S105",
      "uses": [
        "G001",
        "S104"
      ],
      "before": "window20 second divisor:a=4;b=20;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": "S106",
      "uses": [
        "G001",
        "S105"
      ],
      "before": "candidate=20;a divides=否;b divides=是",
      "after": "common multiple=否",
      "relation": "implies",
      "rule_id": "both_divisibility_window_guards",
      "conditions": [],
      "explanation": {
        "zh-Hant": "同一候選必須同時被a與b整除；完整包含0和bound端點，遇零除數也依原整除定義處理。",
        "en": "The same candidate must be divisible by both a and b.Include0 and the bound endpoint,and handle zero divisors using the original divisibility definition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S107",
      "uses": [
        "G001",
        "S106"
      ],
      "before": "window21 first divisor:divide natural 21 by positive 6",
      "after": "21=3·6+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": [
          "21",
          "6",
          "3"
        ],
        "output": "3",
        "label": "window21 first divisor"
      }
    },
    {
      "id": "S108",
      "uses": [
        "G001",
        "S107"
      ],
      "before": "window21 first divisor:a=6;b=21;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": "S109",
      "uses": [
        "G001",
        "S108"
      ],
      "before": "window21 second divisor:divide natural 21 by positive 4",
      "after": "21=5·4+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": [
          "21",
          "4",
          "5"
        ],
        "output": "1",
        "label": "window21 second divisor"
      }
    },
    {
      "id": "S110",
      "uses": [
        "G001",
        "S109"
      ],
      "before": "window21 second divisor:a=4;b=21;quotient=5;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": "S111",
      "uses": [
        "G001",
        "S110"
      ],
      "before": "candidate=21;a divides=否;b divides=否",
      "after": "common multiple=否",
      "relation": "implies",
      "rule_id": "both_divisibility_window_guards",
      "conditions": [],
      "explanation": {
        "zh-Hant": "同一候選必須同時被a與b整除；完整包含0和bound端點，遇零除數也依原整除定義處理。",
        "en": "The same candidate must be divisible by both a and b.Include0 and the bound endpoint,and handle zero divisors using the original divisibility definition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S112",
      "uses": [
        "G001",
        "S111"
      ],
      "before": "window22 first divisor:divide natural 22 by positive 6",
      "after": "22=3·6+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": [
          "22",
          "6",
          "3"
        ],
        "output": "4",
        "label": "window22 first divisor"
      }
    },
    {
      "id": "S113",
      "uses": [
        "G001",
        "S112"
      ],
      "before": "window22 first divisor:a=6;b=22;quotient=3;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": "S114",
      "uses": [
        "G001",
        "S113"
      ],
      "before": "window22 second divisor:divide natural 22 by positive 4",
      "after": "22=5·4+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": [
          "22",
          "4",
          "5"
        ],
        "output": "2",
        "label": "window22 second divisor"
      }
    },
    {
      "id": "S115",
      "uses": [
        "G001",
        "S114"
      ],
      "before": "window22 second divisor:a=4;b=22;quotient=5;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": "S116",
      "uses": [
        "G001",
        "S115"
      ],
      "before": "candidate=22;a divides=否;b divides=否",
      "after": "common multiple=否",
      "relation": "implies",
      "rule_id": "both_divisibility_window_guards",
      "conditions": [],
      "explanation": {
        "zh-Hant": "同一候選必須同時被a與b整除；完整包含0和bound端點，遇零除數也依原整除定義處理。",
        "en": "The same candidate must be divisible by both a and b.Include0 and the bound endpoint,and handle zero divisors using the original divisibility definition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S117",
      "uses": [
        "G001",
        "S116"
      ],
      "before": "window23 first divisor:divide natural 23 by positive 6",
      "after": "23=3·6+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": [
          "23",
          "6",
          "3"
        ],
        "output": "5",
        "label": "window23 first divisor"
      }
    },
    {
      "id": "S118",
      "uses": [
        "G001",
        "S117"
      ],
      "before": "window23 first divisor:a=6;b=23;quotient=3;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": "S119",
      "uses": [
        "G001",
        "S118"
      ],
      "before": "window23 second divisor:divide natural 23 by positive 4",
      "after": "23=5·4+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": [
          "23",
          "4",
          "5"
        ],
        "output": "3",
        "label": "window23 second divisor"
      }
    },
    {
      "id": "S120",
      "uses": [
        "G001",
        "S119"
      ],
      "before": "window23 second divisor:a=4;b=23;quotient=5;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": "S121",
      "uses": [
        "G001",
        "S120"
      ],
      "before": "candidate=23;a divides=否;b divides=否",
      "after": "common multiple=否",
      "relation": "implies",
      "rule_id": "both_divisibility_window_guards",
      "conditions": [],
      "explanation": {
        "zh-Hant": "同一候選必須同時被a與b整除；完整包含0和bound端點，遇零除數也依原整除定義處理。",
        "en": "The same candidate must be divisible by both a and b.Include0 and the bound endpoint,and handle zero divisors using the original divisibility definition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S122",
      "uses": [
        "G001",
        "S121"
      ],
      "before": "window24 first divisor:divide natural 24 by positive 6",
      "after": "24=4·6+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": [
          "24",
          "6",
          "4"
        ],
        "output": "0",
        "label": "window24 first divisor"
      }
    },
    {
      "id": "S123",
      "uses": [
        "G001",
        "S122"
      ],
      "before": "window24 first divisor:a=6;b=24;quotient=4;remainder=0",
      "after": "divides=是;witness/status=4",
      "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": "S124",
      "uses": [
        "G001",
        "S123"
      ],
      "before": "window24 second divisor:divide natural 24 by positive 4",
      "after": "24=6·4+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": [
          "24",
          "4",
          "6"
        ],
        "output": "0",
        "label": "window24 second divisor"
      }
    },
    {
      "id": "S125",
      "uses": [
        "G001",
        "S124"
      ],
      "before": "window24 second divisor:a=4;b=24;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": "S126",
      "uses": [
        "G001",
        "S125"
      ],
      "before": "candidate=24;a divides=是;b divides=是",
      "after": "common multiple=是",
      "relation": "implies",
      "rule_id": "both_divisibility_window_guards",
      "conditions": [],
      "explanation": {
        "zh-Hant": "同一候選必須同時被a與b整除；完整包含0和bound端點，遇零除數也依原整除定義處理。",
        "en": "The same candidate must be divisible by both a and b.Include0 and the bound endpoint,and handle zero divisors using the original divisibility definition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S127",
      "uses": [
        "G001",
        "S126"
      ],
      "before": "window25 first divisor:divide natural 25 by positive 6",
      "after": "25=4·6+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": [
          "25",
          "6",
          "4"
        ],
        "output": "1",
        "label": "window25 first divisor"
      }
    },
    {
      "id": "S128",
      "uses": [
        "G001",
        "S127"
      ],
      "before": "window25 first divisor:a=6;b=25;quotient=4;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": "S129",
      "uses": [
        "G001",
        "S128"
      ],
      "before": "window25 second divisor:divide natural 25 by positive 4",
      "after": "25=6·4+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": [
          "25",
          "4",
          "6"
        ],
        "output": "1",
        "label": "window25 second divisor"
      }
    },
    {
      "id": "S130",
      "uses": [
        "G001",
        "S129"
      ],
      "before": "window25 second divisor:a=4;b=25;quotient=6;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": "S131",
      "uses": [
        "G001",
        "S130"
      ],
      "before": "candidate=25;a divides=否;b divides=否",
      "after": "common multiple=否",
      "relation": "implies",
      "rule_id": "both_divisibility_window_guards",
      "conditions": [],
      "explanation": {
        "zh-Hant": "同一候選必須同時被a與b整除；完整包含0和bound端點，遇零除數也依原整除定義處理。",
        "en": "The same candidate must be divisible by both a and b.Include0 and the bound endpoint,and handle zero divisors using the original divisibility definition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S132",
      "uses": [
        "G001",
        "S131"
      ],
      "before": "window26 first divisor:divide natural 26 by positive 6",
      "after": "26=4·6+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": [
          "26",
          "6",
          "4"
        ],
        "output": "2",
        "label": "window26 first divisor"
      }
    },
    {
      "id": "S133",
      "uses": [
        "G001",
        "S132"
      ],
      "before": "window26 first divisor:a=6;b=26;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": "S134",
      "uses": [
        "G001",
        "S133"
      ],
      "before": "window26 second divisor:divide natural 26 by positive 4",
      "after": "26=6·4+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": [
          "26",
          "4",
          "6"
        ],
        "output": "2",
        "label": "window26 second divisor"
      }
    },
    {
      "id": "S135",
      "uses": [
        "G001",
        "S134"
      ],
      "before": "window26 second divisor:a=4;b=26;quotient=6;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": "S136",
      "uses": [
        "G001",
        "S135"
      ],
      "before": "candidate=26;a divides=否;b divides=否",
      "after": "common multiple=否",
      "relation": "implies",
      "rule_id": "both_divisibility_window_guards",
      "conditions": [],
      "explanation": {
        "zh-Hant": "同一候選必須同時被a與b整除；完整包含0和bound端點，遇零除數也依原整除定義處理。",
        "en": "The same candidate must be divisible by both a and b.Include0 and the bound endpoint,and handle zero divisors using the original divisibility definition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S137",
      "uses": [
        "G001",
        "S136"
      ],
      "before": "window27 first divisor:divide natural 27 by positive 6",
      "after": "27=4·6+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": [
          "27",
          "6",
          "4"
        ],
        "output": "3",
        "label": "window27 first divisor"
      }
    },
    {
      "id": "S138",
      "uses": [
        "G001",
        "S137"
      ],
      "before": "window27 first divisor:a=6;b=27;quotient=4;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": "S139",
      "uses": [
        "G001",
        "S138"
      ],
      "before": "window27 second divisor:divide natural 27 by positive 4",
      "after": "27=6·4+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": [
          "27",
          "4",
          "6"
        ],
        "output": "3",
        "label": "window27 second divisor"
      }
    },
    {
      "id": "S140",
      "uses": [
        "G001",
        "S139"
      ],
      "before": "window27 second divisor:a=4;b=27;quotient=6;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": "S141",
      "uses": [
        "G001",
        "S140"
      ],
      "before": "candidate=27;a divides=否;b divides=否",
      "after": "common multiple=否",
      "relation": "implies",
      "rule_id": "both_divisibility_window_guards",
      "conditions": [],
      "explanation": {
        "zh-Hant": "同一候選必須同時被a與b整除；完整包含0和bound端點，遇零除數也依原整除定義處理。",
        "en": "The same candidate must be divisible by both a and b.Include0 and the bound endpoint,and handle zero divisors using the original divisibility definition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S142",
      "uses": [
        "G001",
        "S141"
      ],
      "before": "window28 first divisor:divide natural 28 by positive 6",
      "after": "28=4·6+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": [
          "28",
          "6",
          "4"
        ],
        "output": "4",
        "label": "window28 first divisor"
      }
    },
    {
      "id": "S143",
      "uses": [
        "G001",
        "S142"
      ],
      "before": "window28 first divisor:a=6;b=28;quotient=4;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": "S144",
      "uses": [
        "G001",
        "S143"
      ],
      "before": "window28 second divisor:divide natural 28 by positive 4",
      "after": "28=7·4+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": [
          "28",
          "4",
          "7"
        ],
        "output": "0",
        "label": "window28 second divisor"
      }
    },
    {
      "id": "S145",
      "uses": [
        "G001",
        "S144"
      ],
      "before": "window28 second divisor:a=4;b=28;quotient=7;remainder=0",
      "after": "divides=是;witness/status=7",
      "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": "S146",
      "uses": [
        "G001",
        "S145"
      ],
      "before": "candidate=28;a divides=否;b divides=是",
      "after": "common multiple=否",
      "relation": "implies",
      "rule_id": "both_divisibility_window_guards",
      "conditions": [],
      "explanation": {
        "zh-Hant": "同一候選必須同時被a與b整除；完整包含0和bound端點，遇零除數也依原整除定義處理。",
        "en": "The same candidate must be divisible by both a and b.Include0 and the bound endpoint,and handle zero divisors using the original divisibility definition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S147",
      "uses": [
        "G001",
        "S146"
      ],
      "before": "window29 first divisor:divide natural 29 by positive 6",
      "after": "29=4·6+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": [
          "29",
          "6",
          "4"
        ],
        "output": "5",
        "label": "window29 first divisor"
      }
    },
    {
      "id": "S148",
      "uses": [
        "G001",
        "S147"
      ],
      "before": "window29 first divisor:a=6;b=29;quotient=4;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": "S149",
      "uses": [
        "G001",
        "S148"
      ],
      "before": "window29 second divisor:divide natural 29 by positive 4",
      "after": "29=7·4+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": [
          "29",
          "4",
          "7"
        ],
        "output": "1",
        "label": "window29 second divisor"
      }
    },
    {
      "id": "S150",
      "uses": [
        "G001",
        "S149"
      ],
      "before": "window29 second divisor:a=4;b=29;quotient=7;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": "S151",
      "uses": [
        "G001",
        "S150"
      ],
      "before": "candidate=29;a divides=否;b divides=否",
      "after": "common multiple=否",
      "relation": "implies",
      "rule_id": "both_divisibility_window_guards",
      "conditions": [],
      "explanation": {
        "zh-Hant": "同一候選必須同時被a與b整除；完整包含0和bound端點，遇零除數也依原整除定義處理。",
        "en": "The same candidate must be divisible by both a and b.Include0 and the bound endpoint,and handle zero divisors using the original divisibility definition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S152",
      "uses": [
        "G001",
        "S151"
      ],
      "before": "window30 first divisor: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": "window30 first divisor"
      }
    },
    {
      "id": "S153",
      "uses": [
        "G001",
        "S152"
      ],
      "before": "window30 first divisor: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": "S154",
      "uses": [
        "G001",
        "S153"
      ],
      "before": "window30 second divisor: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": "window30 second divisor"
      }
    },
    {
      "id": "S155",
      "uses": [
        "G001",
        "S154"
      ],
      "before": "window30 second divisor: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": "S156",
      "uses": [
        "G001",
        "S155"
      ],
      "before": "candidate=30;a divides=是;b divides=否",
      "after": "common multiple=否",
      "relation": "implies",
      "rule_id": "both_divisibility_window_guards",
      "conditions": [],
      "explanation": {
        "zh-Hant": "同一候選必須同時被a與b整除；完整包含0和bound端點，遇零除數也依原整除定義處理。",
        "en": "The same candidate must be divisible by both a and b.Include0 and the bound endpoint,and handle zero divisors using the original divisibility definition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S157",
      "uses": [
        "G001",
        "S156"
      ],
      "before": "window31 first divisor:divide natural 31 by positive 6",
      "after": "31=5·6+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": [
          "31",
          "6",
          "5"
        ],
        "output": "1",
        "label": "window31 first divisor"
      }
    },
    {
      "id": "S158",
      "uses": [
        "G001",
        "S157"
      ],
      "before": "window31 first divisor:a=6;b=31;quotient=5;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": "S159",
      "uses": [
        "G001",
        "S158"
      ],
      "before": "window31 second divisor:divide natural 31 by positive 4",
      "after": "31=7·4+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": [
          "31",
          "4",
          "7"
        ],
        "output": "3",
        "label": "window31 second divisor"
      }
    },
    {
      "id": "S160",
      "uses": [
        "G001",
        "S159"
      ],
      "before": "window31 second divisor:a=4;b=31;quotient=7;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": "S161",
      "uses": [
        "G001",
        "S160"
      ],
      "before": "candidate=31;a divides=否;b divides=否",
      "after": "common multiple=否",
      "relation": "implies",
      "rule_id": "both_divisibility_window_guards",
      "conditions": [],
      "explanation": {
        "zh-Hant": "同一候選必須同時被a與b整除；完整包含0和bound端點，遇零除數也依原整除定義處理。",
        "en": "The same candidate must be divisible by both a and b.Include0 and the bound endpoint,and handle zero divisors using the original divisibility definition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S162",
      "uses": [
        "G001",
        "S161"
      ],
      "before": "window32 first divisor:divide natural 32 by positive 6",
      "after": "32=5·6+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": [
          "32",
          "6",
          "5"
        ],
        "output": "2",
        "label": "window32 first divisor"
      }
    },
    {
      "id": "S163",
      "uses": [
        "G001",
        "S162"
      ],
      "before": "window32 first divisor:a=6;b=32;quotient=5;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": "S164",
      "uses": [
        "G001",
        "S163"
      ],
      "before": "window32 second divisor:divide natural 32 by positive 4",
      "after": "32=8·4+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": [
          "32",
          "4",
          "8"
        ],
        "output": "0",
        "label": "window32 second divisor"
      }
    },
    {
      "id": "S165",
      "uses": [
        "G001",
        "S164"
      ],
      "before": "window32 second divisor:a=4;b=32;quotient=8;remainder=0",
      "after": "divides=是;witness/status=8",
      "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": "S166",
      "uses": [
        "G001",
        "S165"
      ],
      "before": "candidate=32;a divides=否;b divides=是",
      "after": "common multiple=否",
      "relation": "implies",
      "rule_id": "both_divisibility_window_guards",
      "conditions": [],
      "explanation": {
        "zh-Hant": "同一候選必須同時被a與b整除；完整包含0和bound端點，遇零除數也依原整除定義處理。",
        "en": "The same candidate must be divisible by both a and b.Include0 and the bound endpoint,and handle zero divisors using the original divisibility definition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S167",
      "uses": [
        "G001",
        "S166"
      ],
      "before": "window33 first divisor:divide natural 33 by positive 6",
      "after": "33=5·6+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": [
          "33",
          "6",
          "5"
        ],
        "output": "3",
        "label": "window33 first divisor"
      }
    },
    {
      "id": "S168",
      "uses": [
        "G001",
        "S167"
      ],
      "before": "window33 first divisor:a=6;b=33;quotient=5;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": "S169",
      "uses": [
        "G001",
        "S168"
      ],
      "before": "window33 second divisor:divide natural 33 by positive 4",
      "after": "33=8·4+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": [
          "33",
          "4",
          "8"
        ],
        "output": "1",
        "label": "window33 second divisor"
      }
    },
    {
      "id": "S170",
      "uses": [
        "G001",
        "S169"
      ],
      "before": "window33 second divisor:a=4;b=33;quotient=8;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": "S171",
      "uses": [
        "G001",
        "S170"
      ],
      "before": "candidate=33;a divides=否;b divides=否",
      "after": "common multiple=否",
      "relation": "implies",
      "rule_id": "both_divisibility_window_guards",
      "conditions": [],
      "explanation": {
        "zh-Hant": "同一候選必須同時被a與b整除；完整包含0和bound端點，遇零除數也依原整除定義處理。",
        "en": "The same candidate must be divisible by both a and b.Include0 and the bound endpoint,and handle zero divisors using the original divisibility definition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S172",
      "uses": [
        "G001",
        "S171"
      ],
      "before": "window34 first divisor:divide natural 34 by positive 6",
      "after": "34=5·6+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": [
          "34",
          "6",
          "5"
        ],
        "output": "4",
        "label": "window34 first divisor"
      }
    },
    {
      "id": "S173",
      "uses": [
        "G001",
        "S172"
      ],
      "before": "window34 first divisor:a=6;b=34;quotient=5;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": "S174",
      "uses": [
        "G001",
        "S173"
      ],
      "before": "window34 second divisor:divide natural 34 by positive 4",
      "after": "34=8·4+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": [
          "34",
          "4",
          "8"
        ],
        "output": "2",
        "label": "window34 second divisor"
      }
    },
    {
      "id": "S175",
      "uses": [
        "G001",
        "S174"
      ],
      "before": "window34 second divisor:a=4;b=34;quotient=8;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": "S176",
      "uses": [
        "G001",
        "S175"
      ],
      "before": "candidate=34;a divides=否;b divides=否",
      "after": "common multiple=否",
      "relation": "implies",
      "rule_id": "both_divisibility_window_guards",
      "conditions": [],
      "explanation": {
        "zh-Hant": "同一候選必須同時被a與b整除；完整包含0和bound端點，遇零除數也依原整除定義處理。",
        "en": "The same candidate must be divisible by both a and b.Include0 and the bound endpoint,and handle zero divisors using the original divisibility definition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S177",
      "uses": [
        "G001",
        "S176"
      ],
      "before": "window35 first divisor:divide natural 35 by positive 6",
      "after": "35=5·6+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": [
          "35",
          "6",
          "5"
        ],
        "output": "5",
        "label": "window35 first divisor"
      }
    },
    {
      "id": "S178",
      "uses": [
        "G001",
        "S177"
      ],
      "before": "window35 first divisor:a=6;b=35;quotient=5;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": "S179",
      "uses": [
        "G001",
        "S178"
      ],
      "before": "window35 second divisor:divide natural 35 by positive 4",
      "after": "35=8·4+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": [
          "35",
          "4",
          "8"
        ],
        "output": "3",
        "label": "window35 second divisor"
      }
    },
    {
      "id": "S180",
      "uses": [
        "G001",
        "S179"
      ],
      "before": "window35 second divisor:a=4;b=35;quotient=8;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": "S181",
      "uses": [
        "G001",
        "S180"
      ],
      "before": "candidate=35;a divides=否;b divides=否",
      "after": "common multiple=否",
      "relation": "implies",
      "rule_id": "both_divisibility_window_guards",
      "conditions": [],
      "explanation": {
        "zh-Hant": "同一候選必須同時被a與b整除；完整包含0和bound端點，遇零除數也依原整除定義處理。",
        "en": "The same candidate must be divisible by both a and b.Include0 and the bound endpoint,and handle zero divisors using the original divisibility definition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S182",
      "uses": [
        "G001",
        "S181"
      ],
      "before": "window36 first divisor:divide natural 36 by positive 6",
      "after": "36=6·6+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": [
          "36",
          "6",
          "6"
        ],
        "output": "0",
        "label": "window36 first divisor"
      }
    },
    {
      "id": "S183",
      "uses": [
        "G001",
        "S182"
      ],
      "before": "window36 first divisor:a=6;b=36;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": "S184",
      "uses": [
        "G001",
        "S183"
      ],
      "before": "window36 second divisor:divide natural 36 by positive 4",
      "after": "36=9·4+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": [
          "36",
          "4",
          "9"
        ],
        "output": "0",
        "label": "window36 second divisor"
      }
    },
    {
      "id": "S185",
      "uses": [
        "G001",
        "S184"
      ],
      "before": "window36 second divisor:a=4;b=36;quotient=9;remainder=0",
      "after": "divides=是;witness/status=9",
      "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": "S186",
      "uses": [
        "G001",
        "S185"
      ],
      "before": "candidate=36;a divides=是;b divides=是",
      "after": "common multiple=是",
      "relation": "implies",
      "rule_id": "both_divisibility_window_guards",
      "conditions": [],
      "explanation": {
        "zh-Hant": "同一候選必須同時被a與b整除；完整包含0和bound端點，遇零除數也依原整除定義處理。",
        "en": "The same candidate must be divisible by both a and b.Include0 and the bound endpoint,and handle zero divisors using the original divisibility definition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S187",
      "uses": [
        "G001",
        "S186"
      ],
      "before": "window37 first divisor:divide natural 37 by positive 6",
      "after": "37=6·6+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": [
          "37",
          "6",
          "6"
        ],
        "output": "1",
        "label": "window37 first divisor"
      }
    },
    {
      "id": "S188",
      "uses": [
        "G001",
        "S187"
      ],
      "before": "window37 first divisor:a=6;b=37;quotient=6;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": "S189",
      "uses": [
        "G001",
        "S188"
      ],
      "before": "window37 second divisor:divide natural 37 by positive 4",
      "after": "37=9·4+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": [
          "37",
          "4",
          "9"
        ],
        "output": "1",
        "label": "window37 second divisor"
      }
    },
    {
      "id": "S190",
      "uses": [
        "G001",
        "S189"
      ],
      "before": "window37 second divisor:a=4;b=37;quotient=9;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": "S191",
      "uses": [
        "G001",
        "S190"
      ],
      "before": "candidate=37;a divides=否;b divides=否",
      "after": "common multiple=否",
      "relation": "implies",
      "rule_id": "both_divisibility_window_guards",
      "conditions": [],
      "explanation": {
        "zh-Hant": "同一候選必須同時被a與b整除；完整包含0和bound端點，遇零除數也依原整除定義處理。",
        "en": "The same candidate must be divisible by both a and b.Include0 and the bound endpoint,and handle zero divisors using the original divisibility definition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S192",
      "uses": [
        "G001",
        "S191"
      ],
      "before": "window38 first divisor:divide natural 38 by positive 6",
      "after": "38=6·6+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": [
          "38",
          "6",
          "6"
        ],
        "output": "2",
        "label": "window38 first divisor"
      }
    },
    {
      "id": "S193",
      "uses": [
        "G001",
        "S192"
      ],
      "before": "window38 first divisor:a=6;b=38;quotient=6;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": "S194",
      "uses": [
        "G001",
        "S193"
      ],
      "before": "window38 second divisor:divide natural 38 by positive 4",
      "after": "38=9·4+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": [
          "38",
          "4",
          "9"
        ],
        "output": "2",
        "label": "window38 second divisor"
      }
    },
    {
      "id": "S195",
      "uses": [
        "G001",
        "S194"
      ],
      "before": "window38 second divisor:a=4;b=38;quotient=9;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": "S196",
      "uses": [
        "G001",
        "S195"
      ],
      "before": "candidate=38;a divides=否;b divides=否",
      "after": "common multiple=否",
      "relation": "implies",
      "rule_id": "both_divisibility_window_guards",
      "conditions": [],
      "explanation": {
        "zh-Hant": "同一候選必須同時被a與b整除；完整包含0和bound端點，遇零除數也依原整除定義處理。",
        "en": "The same candidate must be divisible by both a and b.Include0 and the bound endpoint,and handle zero divisors using the original divisibility definition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S197",
      "uses": [
        "G001",
        "S196"
      ],
      "before": "window39 first divisor:divide natural 39 by positive 6",
      "after": "39=6·6+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": [
          "39",
          "6",
          "6"
        ],
        "output": "3",
        "label": "window39 first divisor"
      }
    },
    {
      "id": "S198",
      "uses": [
        "G001",
        "S197"
      ],
      "before": "window39 first divisor:a=6;b=39;quotient=6;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": "S199",
      "uses": [
        "G001",
        "S198"
      ],
      "before": "window39 second divisor:divide natural 39 by positive 4",
      "after": "39=9·4+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": [
          "39",
          "4",
          "9"
        ],
        "output": "3",
        "label": "window39 second divisor"
      }
    },
    {
      "id": "S200",
      "uses": [
        "G001",
        "S199"
      ],
      "before": "window39 second divisor:a=4;b=39;quotient=9;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": "S201",
      "uses": [
        "G001",
        "S200"
      ],
      "before": "candidate=39;a divides=否;b divides=否",
      "after": "common multiple=否",
      "relation": "implies",
      "rule_id": "both_divisibility_window_guards",
      "conditions": [],
      "explanation": {
        "zh-Hant": "同一候選必須同時被a與b整除；完整包含0和bound端點，遇零除數也依原整除定義處理。",
        "en": "The same candidate must be divisible by both a and b.Include0 and the bound endpoint,and handle zero divisors using the original divisibility definition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S202",
      "uses": [
        "G001",
        "S201"
      ],
      "before": "window40 first divisor:divide natural 40 by positive 6",
      "after": "40=6·6+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": [
          "40",
          "6",
          "6"
        ],
        "output": "4",
        "label": "window40 first divisor"
      }
    },
    {
      "id": "S203",
      "uses": [
        "G001",
        "S202"
      ],
      "before": "window40 first divisor:a=6;b=40;quotient=6;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": "S204",
      "uses": [
        "G001",
        "S203"
      ],
      "before": "window40 second divisor:divide natural 40 by positive 4",
      "after": "40=10·4+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": [
          "40",
          "4",
          "10"
        ],
        "output": "0",
        "label": "window40 second divisor"
      }
    },
    {
      "id": "S205",
      "uses": [
        "G001",
        "S204"
      ],
      "before": "window40 second divisor:a=4;b=40;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": "S206",
      "uses": [
        "G001",
        "S205"
      ],
      "before": "candidate=40;a divides=否;b divides=是",
      "after": "common multiple=否",
      "relation": "implies",
      "rule_id": "both_divisibility_window_guards",
      "conditions": [],
      "explanation": {
        "zh-Hant": "同一候選必須同時被a與b整除；完整包含0和bound端點，遇零除數也依原整除定義處理。",
        "en": "The same candidate must be divisible by both a and b.Include0 and the bound endpoint,and handle zero divisors using the original divisibility definition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S207",
      "uses": [
        "G001",
        "S206"
      ],
      "before": "window41 first divisor:divide natural 41 by positive 6",
      "after": "41=6·6+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": [
          "41",
          "6",
          "6"
        ],
        "output": "5",
        "label": "window41 first divisor"
      }
    },
    {
      "id": "S208",
      "uses": [
        "G001",
        "S207"
      ],
      "before": "window41 first divisor:a=6;b=41;quotient=6;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": "S209",
      "uses": [
        "G001",
        "S208"
      ],
      "before": "window41 second divisor:divide natural 41 by positive 4",
      "after": "41=10·4+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": [
          "41",
          "4",
          "10"
        ],
        "output": "1",
        "label": "window41 second divisor"
      }
    },
    {
      "id": "S210",
      "uses": [
        "G001",
        "S209"
      ],
      "before": "window41 second divisor:a=4;b=41;quotient=10;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": "S211",
      "uses": [
        "G001",
        "S210"
      ],
      "before": "candidate=41;a divides=否;b divides=否",
      "after": "common multiple=否",
      "relation": "implies",
      "rule_id": "both_divisibility_window_guards",
      "conditions": [],
      "explanation": {
        "zh-Hant": "同一候選必須同時被a與b整除；完整包含0和bound端點，遇零除數也依原整除定義處理。",
        "en": "The same candidate must be divisible by both a and b.Include0 and the bound endpoint,and handle zero divisors using the original divisibility definition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S212",
      "uses": [
        "G001",
        "S211"
      ],
      "before": "window42 first divisor:divide natural 42 by positive 6",
      "after": "42=7·6+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": [
          "42",
          "6",
          "7"
        ],
        "output": "0",
        "label": "window42 first divisor"
      }
    },
    {
      "id": "S213",
      "uses": [
        "G001",
        "S212"
      ],
      "before": "window42 first divisor:a=6;b=42;quotient=7;remainder=0",
      "after": "divides=是;witness/status=7",
      "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": "S214",
      "uses": [
        "G001",
        "S213"
      ],
      "before": "window42 second divisor:divide natural 42 by positive 4",
      "after": "42=10·4+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": [
          "42",
          "4",
          "10"
        ],
        "output": "2",
        "label": "window42 second divisor"
      }
    },
    {
      "id": "S215",
      "uses": [
        "G001",
        "S214"
      ],
      "before": "window42 second divisor:a=4;b=42;quotient=10;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": "S216",
      "uses": [
        "G001",
        "S215"
      ],
      "before": "candidate=42;a divides=是;b divides=否",
      "after": "common multiple=否",
      "relation": "implies",
      "rule_id": "both_divisibility_window_guards",
      "conditions": [],
      "explanation": {
        "zh-Hant": "同一候選必須同時被a與b整除；完整包含0和bound端點，遇零除數也依原整除定義處理。",
        "en": "The same candidate must be divisible by both a and b.Include0 and the bound endpoint,and handle zero divisors using the original divisibility definition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S217",
      "uses": [
        "G001",
        "S216"
      ],
      "before": "window43 first divisor:divide natural 43 by positive 6",
      "after": "43=7·6+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": [
          "43",
          "6",
          "7"
        ],
        "output": "1",
        "label": "window43 first divisor"
      }
    },
    {
      "id": "S218",
      "uses": [
        "G001",
        "S217"
      ],
      "before": "window43 first divisor:a=6;b=43;quotient=7;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": "S219",
      "uses": [
        "G001",
        "S218"
      ],
      "before": "window43 second divisor:divide natural 43 by positive 4",
      "after": "43=10·4+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": [
          "43",
          "4",
          "10"
        ],
        "output": "3",
        "label": "window43 second divisor"
      }
    },
    {
      "id": "S220",
      "uses": [
        "G001",
        "S219"
      ],
      "before": "window43 second divisor:a=4;b=43;quotient=10;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": "S221",
      "uses": [
        "G001",
        "S220"
      ],
      "before": "candidate=43;a divides=否;b divides=否",
      "after": "common multiple=否",
      "relation": "implies",
      "rule_id": "both_divisibility_window_guards",
      "conditions": [],
      "explanation": {
        "zh-Hant": "同一候選必須同時被a與b整除；完整包含0和bound端點，遇零除數也依原整除定義處理。",
        "en": "The same candidate must be divisible by both a and b.Include0 and the bound endpoint,and handle zero divisors using the original divisibility definition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S222",
      "uses": [
        "G001",
        "S221"
      ],
      "before": "window44 first divisor:divide natural 44 by positive 6",
      "after": "44=7·6+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": [
          "44",
          "6",
          "7"
        ],
        "output": "2",
        "label": "window44 first divisor"
      }
    },
    {
      "id": "S223",
      "uses": [
        "G001",
        "S222"
      ],
      "before": "window44 first divisor:a=6;b=44;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": "S224",
      "uses": [
        "G001",
        "S223"
      ],
      "before": "window44 second divisor:divide natural 44 by positive 4",
      "after": "44=11·4+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": [
          "44",
          "4",
          "11"
        ],
        "output": "0",
        "label": "window44 second divisor"
      }
    },
    {
      "id": "S225",
      "uses": [
        "G001",
        "S224"
      ],
      "before": "window44 second divisor:a=4;b=44;quotient=11;remainder=0",
      "after": "divides=是;witness/status=11",
      "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": "S226",
      "uses": [
        "G001",
        "S225"
      ],
      "before": "candidate=44;a divides=否;b divides=是",
      "after": "common multiple=否",
      "relation": "implies",
      "rule_id": "both_divisibility_window_guards",
      "conditions": [],
      "explanation": {
        "zh-Hant": "同一候選必須同時被a與b整除；完整包含0和bound端點，遇零除數也依原整除定義處理。",
        "en": "The same candidate must be divisible by both a and b.Include0 and the bound endpoint,and handle zero divisors using the original divisibility definition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S227",
      "uses": [
        "G001",
        "S226"
      ],
      "before": "window45 first divisor:divide natural 45 by positive 6",
      "after": "45=7·6+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": [
          "45",
          "6",
          "7"
        ],
        "output": "3",
        "label": "window45 first divisor"
      }
    },
    {
      "id": "S228",
      "uses": [
        "G001",
        "S227"
      ],
      "before": "window45 first divisor:a=6;b=45;quotient=7;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": "S229",
      "uses": [
        "G001",
        "S228"
      ],
      "before": "window45 second divisor:divide natural 45 by positive 4",
      "after": "45=11·4+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": [
          "45",
          "4",
          "11"
        ],
        "output": "1",
        "label": "window45 second divisor"
      }
    },
    {
      "id": "S230",
      "uses": [
        "G001",
        "S229"
      ],
      "before": "window45 second divisor:a=4;b=45;quotient=11;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": "S231",
      "uses": [
        "G001",
        "S230"
      ],
      "before": "candidate=45;a divides=否;b divides=否",
      "after": "common multiple=否",
      "relation": "implies",
      "rule_id": "both_divisibility_window_guards",
      "conditions": [],
      "explanation": {
        "zh-Hant": "同一候選必須同時被a與b整除；完整包含0和bound端點，遇零除數也依原整除定義處理。",
        "en": "The same candidate must be divisible by both a and b.Include0 and the bound endpoint,and handle zero divisors using the original divisibility definition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S232",
      "uses": [
        "G001",
        "S231"
      ],
      "before": "window46 first divisor:divide natural 46 by positive 6",
      "after": "46=7·6+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": [
          "46",
          "6",
          "7"
        ],
        "output": "4",
        "label": "window46 first divisor"
      }
    },
    {
      "id": "S233",
      "uses": [
        "G001",
        "S232"
      ],
      "before": "window46 first divisor:a=6;b=46;quotient=7;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": "S234",
      "uses": [
        "G001",
        "S233"
      ],
      "before": "window46 second divisor:divide natural 46 by positive 4",
      "after": "46=11·4+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": [
          "46",
          "4",
          "11"
        ],
        "output": "2",
        "label": "window46 second divisor"
      }
    },
    {
      "id": "S235",
      "uses": [
        "G001",
        "S234"
      ],
      "before": "window46 second divisor:a=4;b=46;quotient=11;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": "S236",
      "uses": [
        "G001",
        "S235"
      ],
      "before": "candidate=46;a divides=否;b divides=否",
      "after": "common multiple=否",
      "relation": "implies",
      "rule_id": "both_divisibility_window_guards",
      "conditions": [],
      "explanation": {
        "zh-Hant": "同一候選必須同時被a與b整除；完整包含0和bound端點，遇零除數也依原整除定義處理。",
        "en": "The same candidate must be divisible by both a and b.Include0 and the bound endpoint,and handle zero divisors using the original divisibility definition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S237",
      "uses": [
        "G001",
        "S236"
      ],
      "before": "window47 first divisor:divide natural 47 by positive 6",
      "after": "47=7·6+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": [
          "47",
          "6",
          "7"
        ],
        "output": "5",
        "label": "window47 first divisor"
      }
    },
    {
      "id": "S238",
      "uses": [
        "G001",
        "S237"
      ],
      "before": "window47 first divisor:a=6;b=47;quotient=7;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": "S239",
      "uses": [
        "G001",
        "S238"
      ],
      "before": "window47 second divisor:divide natural 47 by positive 4",
      "after": "47=11·4+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": [
          "47",
          "4",
          "11"
        ],
        "output": "3",
        "label": "window47 second divisor"
      }
    },
    {
      "id": "S240",
      "uses": [
        "G001",
        "S239"
      ],
      "before": "window47 second divisor:a=4;b=47;quotient=11;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": "S241",
      "uses": [
        "G001",
        "S240"
      ],
      "before": "candidate=47;a divides=否;b divides=否",
      "after": "common multiple=否",
      "relation": "implies",
      "rule_id": "both_divisibility_window_guards",
      "conditions": [],
      "explanation": {
        "zh-Hant": "同一候選必須同時被a與b整除；完整包含0和bound端點，遇零除數也依原整除定義處理。",
        "en": "The same candidate must be divisible by both a and b.Include0 and the bound endpoint,and handle zero divisors using the original divisibility definition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S242",
      "uses": [
        "G001",
        "S241"
      ],
      "before": "window48 first divisor:divide natural 48 by positive 6",
      "after": "48=8·6+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": [
          "48",
          "6",
          "8"
        ],
        "output": "0",
        "label": "window48 first divisor"
      }
    },
    {
      "id": "S243",
      "uses": [
        "G001",
        "S242"
      ],
      "before": "window48 first divisor:a=6;b=48;quotient=8;remainder=0",
      "after": "divides=是;witness/status=8",
      "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": "S244",
      "uses": [
        "G001",
        "S243"
      ],
      "before": "window48 second divisor:divide natural 48 by positive 4",
      "after": "48=12·4+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": [
          "48",
          "4",
          "12"
        ],
        "output": "0",
        "label": "window48 second divisor"
      }
    },
    {
      "id": "S245",
      "uses": [
        "G001",
        "S244"
      ],
      "before": "window48 second divisor:a=4;b=48;quotient=12;remainder=0",
      "after": "divides=是;witness/status=12",
      "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": "S246",
      "uses": [
        "G001",
        "S245"
      ],
      "before": "candidate=48;a divides=是;b divides=是",
      "after": "common multiple=是",
      "relation": "implies",
      "rule_id": "both_divisibility_window_guards",
      "conditions": [],
      "explanation": {
        "zh-Hant": "同一候選必須同時被a與b整除；完整包含0和bound端點，遇零除數也依原整除定義處理。",
        "en": "The same candidate must be divisible by both a and b.Include0 and the bound endpoint,and handle zero divisors using the original divisibility definition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S247",
      "uses": [
        "G001",
        "S246"
      ],
      "before": "window49 first divisor:divide natural 49 by positive 6",
      "after": "49=8·6+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": [
          "49",
          "6",
          "8"
        ],
        "output": "1",
        "label": "window49 first divisor"
      }
    },
    {
      "id": "S248",
      "uses": [
        "G001",
        "S247"
      ],
      "before": "window49 first divisor:a=6;b=49;quotient=8;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": "S249",
      "uses": [
        "G001",
        "S248"
      ],
      "before": "window49 second divisor:divide natural 49 by positive 4",
      "after": "49=12·4+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": [
          "49",
          "4",
          "12"
        ],
        "output": "1",
        "label": "window49 second divisor"
      }
    },
    {
      "id": "S250",
      "uses": [
        "G001",
        "S249"
      ],
      "before": "window49 second divisor:a=4;b=49;quotient=12;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": "S251",
      "uses": [
        "G001",
        "S250"
      ],
      "before": "candidate=49;a divides=否;b divides=否",
      "after": "common multiple=否",
      "relation": "implies",
      "rule_id": "both_divisibility_window_guards",
      "conditions": [],
      "explanation": {
        "zh-Hant": "同一候選必須同時被a與b整除；完整包含0和bound端點，遇零除數也依原整除定義處理。",
        "en": "The same candidate must be divisible by both a and b.Include0 and the bound endpoint,and handle zero divisors using the original divisibility definition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S252",
      "uses": [
        "G001",
        "S251"
      ],
      "before": "window50 first divisor:divide natural 50 by positive 6",
      "after": "50=8·6+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": [
          "50",
          "6",
          "8"
        ],
        "output": "2",
        "label": "window50 first divisor"
      }
    },
    {
      "id": "S253",
      "uses": [
        "G001",
        "S252"
      ],
      "before": "window50 first divisor:a=6;b=50;quotient=8;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": "S254",
      "uses": [
        "G001",
        "S253"
      ],
      "before": "window50 second divisor:divide natural 50 by positive 4",
      "after": "50=12·4+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": [
          "50",
          "4",
          "12"
        ],
        "output": "2",
        "label": "window50 second divisor"
      }
    },
    {
      "id": "S255",
      "uses": [
        "G001",
        "S254"
      ],
      "before": "window50 second divisor:a=4;b=50;quotient=12;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": "S256",
      "uses": [
        "G001",
        "S255"
      ],
      "before": "candidate=50;a divides=否;b divides=否",
      "after": "common multiple=否",
      "relation": "implies",
      "rule_id": "both_divisibility_window_guards",
      "conditions": [],
      "explanation": {
        "zh-Hant": "同一候選必須同時被a與b整除；完整包含0和bound端點，遇零除數也依原整除定義處理。",
        "en": "The same candidate must be divisible by both a and b.Include0 and the bound endpoint,and handle zero divisors using the original divisibility definition."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "result": [
      6,
      4,
      50,
      [
        0,
        12,
        24,
        36,
        48
      ]
    ]
  },
  "conclusion": {
    "canonical_value": "(6, 4, 50, [0, 12, 24, 36, 48])",
    "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": 71,
      "original_line_sha256": "91253abe576feabfa0e6bda1eb0578c8cb53480752f6617d94ed5b974e2901a3",
      "question_compact_sha256": "0d7f06a016245e07167922f0b686d8bdbc67e31964e21bb8566c8c15e7025a6b",
      "english_row_compact_sha256": "4786fc9f848544a4963948d8fc53b12f4607780f41a6c50aaa5b0bac5e8ac599",
      "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": "f9e38b9d5633ffdb7c429208b28b635d1c9f452b9ebd9a7bdc5e89f9d63b0997",
    "original_candidate_line_sha256": "f9e38b9d5633ffdb7c429208b28b635d1c9f452b9ebd9a7bdc5e89f9d63b0997"
  },
  "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": 256,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "d84e40c11febbc1cd871be60b64a5077a549d5094d0aa9bdcfcebefea7b3bdec",
      "arithmetic_steps": 102
    },
    "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."
  }
}