{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261010-advance-20261014-r05-factorial-0075",
  "pilot_number": "R0012-06750",
  "number": 6750,
  "family_id": "qf-density14-factorial-reported-successor-audit",
  "title": {
    "zh-Hant": "第 6750 題：候選階乘報告的兩個見證",
    "en": "Question 6750: Reported Factorials versus Exact Values and Recurrence"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261010-advance-20261014-r05-factorial-0075",
    "batch_id": "batch-20261010-advance-20261014-r05-factorial-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-factorial"
    ],
    "title_zh": "候選階乘報告的兩個見證",
    "stem_zh": "N從0起；BigInt精確階乘；n=24，候選十進字串u=620448401733239439360000、v=15511210043330985984000000。依序答 (真F(n), 真F(n+1), u−F(n), v−(n+1)u, 雙報告皆真是否)。",
    "answer": {
      "kind": "expression",
      "value": "(620448401733239439360000, 15511210043330985984000000, 0, 0, 是)",
      "explanation_zh": "候選階乘雙見證與缺陷=(620448401733239439360000, 15511210043330985984000000, 0, 0, 是)。另外算真階乘，再分開核初值與遞迴缺陷；遞迴自洽也可能錯；候選自報不是證據。有限遞迴核對非排列計數證明，不作浮點值。"
    },
    "difficulty": {
      "band": "upper_secondary",
      "level": 3,
      "skills": [
        "候選階乘報告的兩個見證"
      ]
    },
    "variant": {
      "family_id": "qf-density14-factorial-reported-successor-audit",
      "seed": 202610145075,
      "parameters": {
        "n": 24,
        "u": "620448401733239439360000",
        "v": "15511210043330985984000000",
        "domain": "finite_N_factorial_protocol",
        "operation": "density14_factorial_reported_successor_audit",
        "semantic_template": "qf-density14-factorial-reported-successor-audit/[24,\"620448401733239439360000\",\"15511210043330985984000000\"]"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_integer_rational_arithmetic",
        "producer_finite_enumeration",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-10",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261010-advance-20261014-r05-factorial-0075",
    "batch_id": "batch-20261010-advance-20261014-r05-factorial-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "ffd763c17e6c9d2fdfdd3e7e4b2098f6abcd48f06c79515cddc82d48c2bfa604",
      "family_id": "qf-density14-factorial-reported-successor-audit",
      "seed": 202610145075,
      "semantic_template": "qf-density14-factorial-reported-successor-audit/[24,\"620448401733239439360000\",\"15511210043330985984000000\"]"
    },
    "fields": {
      "title": "Reported Factorials versus Exact Values and Recurrence",
      "stem": "N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=24; candidate decimal integer strings u=620448401733239439360000, v=15511210043330985984000000. Give (actual F(n), actual F(n+1), u−F(n), v−(n+1)u, are both reports correct?).",
      "answer": {
        "kind": "expression",
        "value": "(620448401733239439360000, 15511210043330985984000000, 0, 0, Yes)",
        "explanation": "Compute true factorials independently before checking the base-value and recurrence defects; recurrence-consistent self-reports can still be wrong and are not Evidence. The exact result, retaining the requested order, is (620448401733239439360000, 15511210043330985984000000, 0, 0, Yes)."
      },
      "skills": [
        "Reported Factorials versus Exact Values and Recurrence"
      ],
      "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": "N從0起；BigInt精確階乘；n=24，候選十進字串u=620448401733239439360000、v=15511210043330985984000000。依序答 (真F(n), 真F(n+1), u−F(n), v−(n+1)u, 雙報告皆真是否)。",
        "en": "N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=24; candidate decimal integer strings u=620448401733239439360000, v=15511210043330985984000000. Give (actual F(n), actual F(n+1), u−F(n), v−(n+1)u, are both reports correct?)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "N includes0;F(0)=1;F(j)=j·F(j−1)",
      "after": "exact integer products,not floating values or a permutation proof",
      "relation": "implies",
      "rule_id": "exact_factorial_protocol",
      "conditions": [],
      "explanation": {
        "zh-Hant": "階乘使用空乘積1及逐因子整數乘積，所有大數精確保留；有限遞迴核對不冒充一般排列計數證明。",
        "en": "Use empty product1 and integer factor multiplication,retaining large values exactly.This finite recurrence audit is not a general proof of permutation counting."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "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": "actual factorial:start at F(0)",
      "after": "F(0)=1",
      "relation": "implies",
      "rule_id": "factorial_empty_product_base",
      "conditions": [],
      "explanation": {
        "zh-Hant": "0個因子的空乘積是1，不是0；這是後繼遞推的起點。",
        "en": "The product of zero factors is1,not0;this is the successor recurrence's base."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "actual factorial through factor1 := mul(1, 1)",
      "after": "actual factorial through factor1 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "把新正整數因子乘入前一階乘值，保留全部中間整數。",
        "en": "Multiply the preceding factorial by the new positive integer factor,retaining every exact intermediate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 actual factorial through factor1，不以小數近似替代。",
          "en": "Recompute actual factorial through factor1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "1"
        ],
        "output": "1",
        "label": "actual factorial through factor1"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "actual factorial through factor2 := mul(1, 2)",
      "after": "actual factorial through factor2 = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "把新正整數因子乘入前一階乘值，保留全部中間整數。",
        "en": "Multiply the preceding factorial by the new positive integer factor,retaining every exact intermediate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 actual factorial through factor2，不以小數近似替代。",
          "en": "Recompute actual factorial through factor2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "2"
        ],
        "output": "2",
        "label": "actual factorial through factor2"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "actual factorial through factor3 := mul(2, 3)",
      "after": "actual factorial through factor3 = 6",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "把新正整數因子乘入前一階乘值，保留全部中間整數。",
        "en": "Multiply the preceding factorial by the new positive integer factor,retaining every exact intermediate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 actual factorial through factor3，不以小數近似替代。",
          "en": "Recompute actual factorial through factor3 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "3"
        ],
        "output": "6",
        "label": "actual factorial through factor3"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "actual factorial through factor4 := mul(6, 4)",
      "after": "actual factorial through factor4 = 24",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "把新正整數因子乘入前一階乘值，保留全部中間整數。",
        "en": "Multiply the preceding factorial by the new positive integer factor,retaining every exact intermediate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 actual factorial through factor4，不以小數近似替代。",
          "en": "Recompute actual factorial through factor4 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "6",
          "4"
        ],
        "output": "24",
        "label": "actual factorial through factor4"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "actual factorial through factor5 := mul(24, 5)",
      "after": "actual factorial through factor5 = 120",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "把新正整數因子乘入前一階乘值，保留全部中間整數。",
        "en": "Multiply the preceding factorial by the new positive integer factor,retaining every exact intermediate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 actual factorial through factor5，不以小數近似替代。",
          "en": "Recompute actual factorial through factor5 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "24",
          "5"
        ],
        "output": "120",
        "label": "actual factorial through factor5"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "actual factorial through factor6 := mul(120, 6)",
      "after": "actual factorial through factor6 = 720",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "把新正整數因子乘入前一階乘值，保留全部中間整數。",
        "en": "Multiply the preceding factorial by the new positive integer factor,retaining every exact intermediate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 actual factorial through factor6，不以小數近似替代。",
          "en": "Recompute actual factorial through factor6 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "120",
          "6"
        ],
        "output": "720",
        "label": "actual factorial through factor6"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "actual factorial through factor7 := mul(720, 7)",
      "after": "actual factorial through factor7 = 5040",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "把新正整數因子乘入前一階乘值，保留全部中間整數。",
        "en": "Multiply the preceding factorial by the new positive integer factor,retaining every exact intermediate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 actual factorial through factor7，不以小數近似替代。",
          "en": "Recompute actual factorial through factor7 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "720",
          "7"
        ],
        "output": "5040",
        "label": "actual factorial through factor7"
      }
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "actual factorial through factor8 := mul(5040, 8)",
      "after": "actual factorial through factor8 = 40320",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "把新正整數因子乘入前一階乘值，保留全部中間整數。",
        "en": "Multiply the preceding factorial by the new positive integer factor,retaining every exact intermediate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 actual factorial through factor8，不以小數近似替代。",
          "en": "Recompute actual factorial through factor8 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "5040",
          "8"
        ],
        "output": "40320",
        "label": "actual factorial through factor8"
      }
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "actual factorial through factor9 := mul(40320, 9)",
      "after": "actual factorial through factor9 = 362880",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "把新正整數因子乘入前一階乘值，保留全部中間整數。",
        "en": "Multiply the preceding factorial by the new positive integer factor,retaining every exact intermediate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 actual factorial through factor9，不以小數近似替代。",
          "en": "Recompute actual factorial through factor9 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "40320",
          "9"
        ],
        "output": "362880",
        "label": "actual factorial through factor9"
      }
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "actual factorial through factor10 := mul(362880, 10)",
      "after": "actual factorial through factor10 = 3628800",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "把新正整數因子乘入前一階乘值，保留全部中間整數。",
        "en": "Multiply the preceding factorial by the new positive integer factor,retaining every exact intermediate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 actual factorial through factor10，不以小數近似替代。",
          "en": "Recompute actual factorial through factor10 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "362880",
          "10"
        ],
        "output": "3628800",
        "label": "actual factorial through factor10"
      }
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "actual factorial through factor11 := mul(3628800, 11)",
      "after": "actual factorial through factor11 = 39916800",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "把新正整數因子乘入前一階乘值，保留全部中間整數。",
        "en": "Multiply the preceding factorial by the new positive integer factor,retaining every exact intermediate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 actual factorial through factor11，不以小數近似替代。",
          "en": "Recompute actual factorial through factor11 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "3628800",
          "11"
        ],
        "output": "39916800",
        "label": "actual factorial through factor11"
      }
    },
    {
      "id": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "actual factorial through factor12 := mul(39916800, 12)",
      "after": "actual factorial through factor12 = 479001600",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "把新正整數因子乘入前一階乘值，保留全部中間整數。",
        "en": "Multiply the preceding factorial by the new positive integer factor,retaining every exact intermediate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 actual factorial through factor12，不以小數近似替代。",
          "en": "Recompute actual factorial through factor12 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "39916800",
          "12"
        ],
        "output": "479001600",
        "label": "actual factorial through factor12"
      }
    },
    {
      "id": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "actual factorial through factor13 := mul(479001600, 13)",
      "after": "actual factorial through factor13 = 6227020800",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "把新正整數因子乘入前一階乘值，保留全部中間整數。",
        "en": "Multiply the preceding factorial by the new positive integer factor,retaining every exact intermediate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 actual factorial through factor13，不以小數近似替代。",
          "en": "Recompute actual factorial through factor13 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "479001600",
          "13"
        ],
        "output": "6227020800",
        "label": "actual factorial through factor13"
      }
    },
    {
      "id": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "actual factorial through factor14 := mul(6227020800, 14)",
      "after": "actual factorial through factor14 = 87178291200",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "把新正整數因子乘入前一階乘值，保留全部中間整數。",
        "en": "Multiply the preceding factorial by the new positive integer factor,retaining every exact intermediate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 actual factorial through factor14，不以小數近似替代。",
          "en": "Recompute actual factorial through factor14 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "6227020800",
          "14"
        ],
        "output": "87178291200",
        "label": "actual factorial through factor14"
      }
    },
    {
      "id": "S017",
      "uses": [
        "G001",
        "S016"
      ],
      "before": "actual factorial through factor15 := mul(87178291200, 15)",
      "after": "actual factorial through factor15 = 1307674368000",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "把新正整數因子乘入前一階乘值，保留全部中間整數。",
        "en": "Multiply the preceding factorial by the new positive integer factor,retaining every exact intermediate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 actual factorial through factor15，不以小數近似替代。",
          "en": "Recompute actual factorial through factor15 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "87178291200",
          "15"
        ],
        "output": "1307674368000",
        "label": "actual factorial through factor15"
      }
    },
    {
      "id": "S018",
      "uses": [
        "G001",
        "S017"
      ],
      "before": "actual factorial through factor16 := mul(1307674368000, 16)",
      "after": "actual factorial through factor16 = 20922789888000",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "把新正整數因子乘入前一階乘值，保留全部中間整數。",
        "en": "Multiply the preceding factorial by the new positive integer factor,retaining every exact intermediate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 actual factorial through factor16，不以小數近似替代。",
          "en": "Recompute actual factorial through factor16 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1307674368000",
          "16"
        ],
        "output": "20922789888000",
        "label": "actual factorial through factor16"
      }
    },
    {
      "id": "S019",
      "uses": [
        "G001",
        "S018"
      ],
      "before": "actual factorial through factor17 := mul(20922789888000, 17)",
      "after": "actual factorial through factor17 = 355687428096000",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "把新正整數因子乘入前一階乘值，保留全部中間整數。",
        "en": "Multiply the preceding factorial by the new positive integer factor,retaining every exact intermediate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 actual factorial through factor17，不以小數近似替代。",
          "en": "Recompute actual factorial through factor17 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "20922789888000",
          "17"
        ],
        "output": "355687428096000",
        "label": "actual factorial through factor17"
      }
    },
    {
      "id": "S020",
      "uses": [
        "G001",
        "S019"
      ],
      "before": "actual factorial through factor18 := mul(355687428096000, 18)",
      "after": "actual factorial through factor18 = 6402373705728000",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "把新正整數因子乘入前一階乘值，保留全部中間整數。",
        "en": "Multiply the preceding factorial by the new positive integer factor,retaining every exact intermediate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 actual factorial through factor18，不以小數近似替代。",
          "en": "Recompute actual factorial through factor18 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "355687428096000",
          "18"
        ],
        "output": "6402373705728000",
        "label": "actual factorial through factor18"
      }
    },
    {
      "id": "S021",
      "uses": [
        "G001",
        "S020"
      ],
      "before": "actual factorial through factor19 := mul(6402373705728000, 19)",
      "after": "actual factorial through factor19 = 121645100408832000",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "把新正整數因子乘入前一階乘值，保留全部中間整數。",
        "en": "Multiply the preceding factorial by the new positive integer factor,retaining every exact intermediate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 actual factorial through factor19，不以小數近似替代。",
          "en": "Recompute actual factorial through factor19 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "6402373705728000",
          "19"
        ],
        "output": "121645100408832000",
        "label": "actual factorial through factor19"
      }
    },
    {
      "id": "S022",
      "uses": [
        "G001",
        "S021"
      ],
      "before": "actual factorial through factor20 := mul(121645100408832000, 20)",
      "after": "actual factorial through factor20 = 2432902008176640000",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "把新正整數因子乘入前一階乘值，保留全部中間整數。",
        "en": "Multiply the preceding factorial by the new positive integer factor,retaining every exact intermediate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 actual factorial through factor20，不以小數近似替代。",
          "en": "Recompute actual factorial through factor20 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "121645100408832000",
          "20"
        ],
        "output": "2432902008176640000",
        "label": "actual factorial through factor20"
      }
    },
    {
      "id": "S023",
      "uses": [
        "G001",
        "S022"
      ],
      "before": "actual factorial through factor21 := mul(2432902008176640000, 21)",
      "after": "actual factorial through factor21 = 51090942171709440000",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "把新正整數因子乘入前一階乘值，保留全部中間整數。",
        "en": "Multiply the preceding factorial by the new positive integer factor,retaining every exact intermediate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 actual factorial through factor21，不以小數近似替代。",
          "en": "Recompute actual factorial through factor21 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2432902008176640000",
          "21"
        ],
        "output": "51090942171709440000",
        "label": "actual factorial through factor21"
      }
    },
    {
      "id": "S024",
      "uses": [
        "G001",
        "S023"
      ],
      "before": "actual factorial through factor22 := mul(51090942171709440000, 22)",
      "after": "actual factorial through factor22 = 1124000727777607680000",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "把新正整數因子乘入前一階乘值，保留全部中間整數。",
        "en": "Multiply the preceding factorial by the new positive integer factor,retaining every exact intermediate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 actual factorial through factor22，不以小數近似替代。",
          "en": "Recompute actual factorial through factor22 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "51090942171709440000",
          "22"
        ],
        "output": "1124000727777607680000",
        "label": "actual factorial through factor22"
      }
    },
    {
      "id": "S025",
      "uses": [
        "G001",
        "S024"
      ],
      "before": "actual factorial through factor23 := mul(1124000727777607680000, 23)",
      "after": "actual factorial through factor23 = 25852016738884976640000",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "把新正整數因子乘入前一階乘值，保留全部中間整數。",
        "en": "Multiply the preceding factorial by the new positive integer factor,retaining every exact intermediate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 actual factorial through factor23，不以小數近似替代。",
          "en": "Recompute actual factorial through factor23 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1124000727777607680000",
          "23"
        ],
        "output": "25852016738884976640000",
        "label": "actual factorial through factor23"
      }
    },
    {
      "id": "S026",
      "uses": [
        "G001",
        "S025"
      ],
      "before": "actual factorial through factor24 := mul(25852016738884976640000, 24)",
      "after": "actual factorial through factor24 = 620448401733239439360000",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "把新正整數因子乘入前一階乘值，保留全部中間整數。",
        "en": "Multiply the preceding factorial by the new positive integer factor,retaining every exact intermediate."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 actual factorial through factor24，不以小數近似替代。",
          "en": "Recompute actual factorial through factor24 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25852016738884976640000",
          "24"
        ],
        "output": "620448401733239439360000",
        "label": "actual factorial through factor24"
      }
    },
    {
      "id": "S027",
      "uses": [
        "G001",
        "S026"
      ],
      "before": "true successor factorial := mul(25, 620448401733239439360000)",
      "after": "true successor factorial = 15511210043330985984000000",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依F(n+1)=(n+1)F(n)求下一值。",
        "en": "Compute F(n+1)=(n+1)F(n)."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 true successor factorial，不以小數近似替代。",
          "en": "Recompute true successor factorial with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "620448401733239439360000"
        ],
        "output": "15511210043330985984000000",
        "label": "true successor factorial"
      }
    },
    {
      "id": "S028",
      "uses": [
        "G001",
        "S027"
      ],
      "before": "reported u minus true factorial := sub(620448401733239439360000, 620448401733239439360000)",
      "after": "reported u minus true factorial = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "自報u先與另算的真階乘比較，不能只看兩個報告互相自洽。",
        "en": "Compare reported u with the independently computed true factorial,not merely consistency between the reports."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 reported u minus true factorial，不以小數近似替代。",
          "en": "Recompute reported u minus true factorial with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "620448401733239439360000",
          "620448401733239439360000"
        ],
        "output": "0",
        "label": "reported u minus true factorial"
      }
    },
    {
      "id": "S029",
      "uses": [
        "G001",
        "S028"
      ],
      "before": "successor predicted from reported u := mul(25, 620448401733239439360000)",
      "after": "successor predicted from reported u = 15511210043330985984000000",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "由自報u形成候選後繼，與真後繼分列。",
        "en": "Form the candidate successor from reported u,separate from the true successor."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 successor predicted from reported u，不以小數近似替代。",
          "en": "Recompute successor predicted from reported u with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "25",
          "620448401733239439360000"
        ],
        "output": "15511210043330985984000000",
        "label": "successor predicted from reported u"
      }
    },
    {
      "id": "S030",
      "uses": [
        "G001",
        "S029"
      ],
      "before": "reported successor recurrence defect := sub(15511210043330985984000000, 15511210043330985984000000)",
      "after": "reported successor recurrence defect = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "遞迴缺陷為v−(n+1)u，可能0但兩報告仍一起錯。",
        "en": "The recurrence defect is v−(n+1)u;it may be0 while both reports are wrong."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 reported successor recurrence defect，不以小數近似替代。",
          "en": "Recompute reported successor recurrence defect with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "15511210043330985984000000",
          "15511210043330985984000000"
        ],
        "output": "0",
        "label": "reported successor recurrence defect"
      }
    },
    {
      "id": "S031",
      "uses": [
        "G001",
        "S030"
      ],
      "before": "true values=(620448401733239439360000,15511210043330985984000000);reports=(620448401733239439360000,15511210043330985984000000);defects=(0,0)",
      "after": "both reports true=是",
      "relation": "implies",
      "rule_id": "factorial_report_requires_true_anchor",
      "conditions": [],
      "explanation": {
        "zh-Hant": "正確資格同時核真F(n)及真F(n+1)；只有自報遞迴缺陷0不是證據，原始十進字串也保持不變。",
        "en": "Check both true F(n) and true F(n+1);a zero self-reported recurrence defect is not evidence.The original decimal strings remain retained."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "result": [
      "620448401733239439360000",
      "15511210043330985984000000",
      "0",
      "0",
      true
    ]
  },
  "conclusion": {
    "canonical_value": "(620448401733239439360000, 15511210043330985984000000, 0, 0, 是)",
    "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-20261010-advance-20261014-r05-factorial-foundations",
      "batch_path": "public/data/questions/batches/batch-20261010-advance-20261014-r05-factorial-foundations/questions.jsonl",
      "batch_sha256": "50f758b067a5db2003333079c1011c57924774777afe1b86d27959cfc0ad5860",
      "original_line": 75,
      "original_line_sha256": "a24fd10fe47067fbcacd343fe93e82adad1cd6a537018ccd58968abd8ce988f6",
      "question_compact_sha256": "ffd763c17e6c9d2fdfdd3e7e4b2098f6abcd48f06c79515cddc82d48c2bfa604",
      "english_row_compact_sha256": "99325ea87c71cec8697c6dfc50cc9d759843b099cb32dce34da66a90f45865be",
      "english_manifest_path": "public/data/i18n/editions/batch-20261010-advance-20261014-r05-factorial-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261010-advance-20261014-r05-factorial-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-factorial": {
        "sha256": "fc2d013658e47191e660f97c927ceeec97905dd4fc520329d3df326b343a4b8f",
        "english_sha256": "8edff46ffb7ebf142e860dc8cf9cd8e40e799c0039337605934357486ae568c3",
        "original_path": "public/data/mko/factorial.json",
        "english_path": "public/data/i18n/editions/mko-factorial.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "c5c612ca91797c328f4489939ec6f7d885385e63ea21b334c09c421f17933d99",
    "original_candidate_sha256": "912e412d363fb5975782896c00f8beb0628bf23487f15d5bcfa8fc75cc2f90d7",
    "original_candidate_line_sha256": "912e412d363fb5975782896c00f8beb0628bf23487f15d5bcfa8fc75cc2f90d7"
  },
  "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-p13",
    "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": "90dbc494b2d34ef51f412dcff63cfcc1a2215d038bff71f38f5dded4c64faf00",
      "question_first_witness_sha256": "b72a3b8bcf10f584396e0dd5198c657b0934a71864e5a3fa6321d7a613aef9f9"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 31,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "a89f2221129e292ffae317ba8a7bea4d4b1f8f0602d69bb8add0aea4cf9766a8",
      "arithmetic_steps": 28
    },
    "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."
  }
}