{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261010-advance-20261014-r05-factorial-0087",
  "pilot_number": "R0012-06762",
  "number": 6762,
  "family_id": "qf-density14-factorial-single-factor-omission",
  "title": {
    "zh-Hant": "第 6762 題：省去一次因子的乘積",
    "en": "Question 6762: Omitting One Existing Factor and Multiplying Back"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261010-advance-20261014-r05-factorial-0087",
    "batch_id": "batch-20261010-advance-20261014-r05-factorial-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-factorial"
    ],
    "title_zh": "省去一次因子的乘積",
    "stem_zh": "N從0起；BigInt精確階乘；n=13、k=12∈[1,n]。依序求 (n!, 省去一次k的乘積n!/k, k回乘後減n!)。",
    "answer": {
      "kind": "expression",
      "value": "(6227020800, 518918400, 0)",
      "explanation_zh": "省去單因子乘積與回乘=(6227020800, 518918400, 0)。完整乘積另算，省因子乘積跳過指定k一次，再精確回乘。有限遞迴核對非排列計數證明，不作浮點值。"
    },
    "difficulty": {
      "band": "upper_secondary",
      "level": 2,
      "skills": [
        "省去一次因子的乘積"
      ]
    },
    "variant": {
      "family_id": "qf-density14-factorial-single-factor-omission",
      "seed": 202610145087,
      "parameters": {
        "n": 13,
        "k": 12,
        "domain": "finite_N_factorial_protocol",
        "operation": "density14_factorial_single_factor_omission",
        "semantic_template": "qf-density14-factorial-single-factor-omission/[13,12]"
      }
    },
    "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-0087",
    "batch_id": "batch-20261010-advance-20261014-r05-factorial-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "2f0ca7c97845560747dba7139541c635676cca08a4558637424ef79e16cc660c",
      "family_id": "qf-density14-factorial-single-factor-omission",
      "seed": 202610145087,
      "semantic_template": "qf-density14-factorial-single-factor-omission/[13,12]"
    },
    "fields": {
      "title": "Omitting One Existing Factor and Multiplying Back",
      "stem": "N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=13, k=12∈[1,n]. Give (n!, product with k omitted once n!/k, k times that product minus n!).",
      "answer": {
        "kind": "expression",
        "value": "(6227020800, 518918400, 0)",
        "explanation": "Compute the full product independently, omit the specified factor once, then multiply back exactly; no floating-point approximation is used. The exact result, retaining the requested order, is (6227020800, 518918400, 0)."
      },
      "skills": [
        "Omitting One Existing Factor and Multiplying Back"
      ],
      "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=13、k=12∈[1,n]。依序求 (n!, 省去一次k的乘積n!/k, k回乘後減n!)。",
        "en": "N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=13, k=12∈[1,n]. Give (n!, product with k omitted once n!/k, k times that product minus n!)."
      }
    }
  ],
  "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": "single-factor-omitted product := div(6227020800, 12)",
      "after": "single-factor-omitted product = 518918400",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "k在1至n因子列中恰出現一次且非零，除去這一因子即n!/k。",
        "en": "The nonzero k occurs exactly once in the1,...,n factor list;removing that factor gives n!/k."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 single-factor-omitted product，不以小數近似替代。",
          "en": "Recompute single-factor-omitted product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "6227020800",
          "12"
        ],
        "output": "518918400",
        "label": "single-factor-omitted product"
      }
    },
    {
      "id": "S017",
      "uses": [
        "G001",
        "S016"
      ],
      "before": "restore omitted factor := mul(12, 518918400)",
      "after": "restore omitted factor = 6227020800",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "把同一k乘回，核對完整乘積。",
        "en": "Multiply the same k back to audit the complete product."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 restore omitted factor，不以小數近似替代。",
          "en": "Recompute restore omitted factor with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "12",
          "518918400"
        ],
        "output": "6227020800",
        "label": "restore omitted factor"
      }
    },
    {
      "id": "S018",
      "uses": [
        "G001",
        "S017"
      ],
      "before": "restored product minus factorial := sub(6227020800, 6227020800)",
      "after": "restored product minus factorial = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "回乘值減真階乘應為0。",
        "en": "Restored product minus the true factorial must be0."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 restored product minus factorial，不以小數近似替代。",
          "en": "Recompute restored product minus factorial with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "6227020800",
          "6227020800"
        ],
        "output": "0",
        "label": "restored product minus factorial"
      }
    },
    {
      "id": "S019",
      "uses": [
        "G001",
        "S018"
      ],
      "before": "removed factor k=12;full n!=6227020800",
      "after": "omitted product=518918400;restoration residual=0",
      "relation": "implies",
      "rule_id": "single_factor_removal_reverse_audit",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只省指定因子一次，不是把所有k的倍數因子一起省掉；以正整數除法和回乘雙向核對。",
        "en": "Omit the designated factor once,not every factor divisible by k;check by exact positive-integer division and reverse multiplication."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "result": [
      "6227020800",
      "518918400",
      "0"
    ]
  },
  "conclusion": {
    "canonical_value": "(6227020800, 518918400, 0)",
    "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": 87,
      "original_line_sha256": "58c82646111d5c1585d620184211b877ba1a45afa7f988b6a8e9b3405a63d505",
      "question_compact_sha256": "2f0ca7c97845560747dba7139541c635676cca08a4558637424ef79e16cc660c",
      "english_row_compact_sha256": "72f595e8000fb601ab0bb6cd47160cf412ee139311ff930ffbfbd7cf844ea8c4",
      "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": "e29924cecb4007e8e1585e2421b267d3bb9f14360eaaa40253280a921b9f66c3",
    "original_candidate_line_sha256": "e29924cecb4007e8e1585e2421b267d3bb9f14360eaaa40253280a921b9f66c3"
  },
  "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": 19,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "c7b643ac0983298197b3e64c3e36c1e42e2ad9ab7bb1cae23213a5343fadc4c5",
      "arithmetic_steps": 16
    },
    "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."
  }
}