{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20260926-factorial-successor-recurrence-0011",
  "pilot_number": "R0062-31886",
  "number": 31886,
  "family_id": "qf-factorial-successor-recurrence",
  "title": {
    "zh-Hant": "第 31886 題：後繼階乘遞迴",
    "en": "Question 31886: Successor Recurrence for Factorial"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20260926-factorial-successor-recurrence-0011",
    "batch_id": "batch-20260926-factorial-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-factorial"
    ],
    "title_zh": "後繼階乘遞迴",
    "stem_zh": "已知 10!=3628800。用 (n+1)!=(n+1)×n! 求 11!。",
    "answer": {
      "kind": "short_text",
      "value": "39916800",
      "explanation_zh": "11×3628800=39916800，所以 11!=39916800。"
    },
    "difficulty": {
      "band": "lower_secondary",
      "level": 3,
      "skills": [
        "階乘遞迴",
        "BigInt"
      ]
    },
    "variant": {
      "family_id": "qf-factorial-successor-recurrence",
      "seed": 20261037,
      "parameters": {
        "n": 10,
        "n_factorial": "3628800",
        "operation": "successor_recurrence",
        "semantic_template": "qf-factorial-successor-recurrence/{\"n\":10,\"n_factorial\":\"3628800\",\"operation\":\"successor_recurrence\"}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "schema_pending",
        "independent_recomputation_pending"
      ],
      "warning_zh": "本題為 candidate 練習；有限精確重算不是形式證明或人工審查。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "GPT-5.6 Terra",
      "generated_at": "2026-09-26",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20260926-factorial-successor-recurrence-0011",
    "batch_id": "batch-20260926-factorial-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "a60143841fca9cc3da4a76e50fc60326270d608637c1a8920c1020a8270a1622",
      "family_id": "qf-factorial-successor-recurrence",
      "seed": 20261037,
      "semantic_template": "qf-factorial-successor-recurrence/{\"n\":10,\"n_factorial\":\"3628800\",\"operation\":\"successor_recurrence\"}"
    },
    "fields": {
      "title": "Successor Recurrence for Factorial",
      "stem": "Given 10!=3628800, use (n+1)!=(n+1)×n! to find 11!.",
      "answer": {
        "kind": "short_text",
        "value": "39916800",
        "explanation": "11×3628800=39916800, so 11!=39916800."
      },
      "skills": [
        "factorial recurrence",
        "BigInt"
      ],
      "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": "已知 10!=3628800。用 (n+1)!=(n+1)×n! 求 11!。",
        "en": "Given 10!=3628800, use (n+1)!=(n+1)×n! to find 11!."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source roles = [qf-factorial-successor-recurrence, no-domain-field, successor_recurrence, no-form-field]",
      "after": "derived = [[exact_integer_STRING]]",
      "relation": "implies",
      "rule_id": "R62_EXACT_source_keys_domain_operation_form_AND_output_types",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按原bank核欄位：BASE方程用form及論域，BASE事件／補集無operation，階乘無domain；ext18用local-hyphen，其他legacy用underscore。輸出型別不統一成tuple或fraction。",
        "en": "Check THIS bank interface:BASE equations use form/domain;BASE events/complement have no operation;factorials have no domain.ext18 uses local hyphens,other legacy operations underscores.Do not force every answer into a tuple or fraction."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向代回原輸入、欄位角色、論域及本步條件；一致性不表示結論能唯一恢復前提。有限核對、獨立審查、形式證明與發布分開標記。",
          "en": "Reverse-check the ORIGINAL inputs,field roles,domain and premises.Consistency does not imply unique recovery of premises.Keep finite checking,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "Factorial requested n ORIGINAL factorial factor2 := mul(1, 2)",
      "after": "Factorial requested n ORIGINAL factorial factor2 = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "使用原操作數與明示次序作精確整數／有理計算，不用浮點近似。輸入界依題族：不把衍生值重驗成raw輸入，也不給BASE題庫另加SAFE或n64界。",
        "en": "Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Factorial requested n ORIGINAL factorial factor2，不以小數近似替代。",
          "en": "Recompute Factorial requested n ORIGINAL factorial factor2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "2"
        ],
        "output": "2",
        "label": "Factorial requested n ORIGINAL factorial factor2"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "Factorial requested n ORIGINAL factorial factor3 := mul(2, 3)",
      "after": "Factorial requested n ORIGINAL factorial factor3 = 6",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "使用原操作數與明示次序作精確整數／有理計算，不用浮點近似。輸入界依題族：不把衍生值重驗成raw輸入，也不給BASE題庫另加SAFE或n64界。",
        "en": "Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Factorial requested n ORIGINAL factorial factor3，不以小數近似替代。",
          "en": "Recompute Factorial requested n ORIGINAL factorial factor3 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "3"
        ],
        "output": "6",
        "label": "Factorial requested n ORIGINAL factorial factor3"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "Factorial requested n ORIGINAL factorial factor4 := mul(6, 4)",
      "after": "Factorial requested n ORIGINAL factorial factor4 = 24",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "使用原操作數與明示次序作精確整數／有理計算，不用浮點近似。輸入界依題族：不把衍生值重驗成raw輸入，也不給BASE題庫另加SAFE或n64界。",
        "en": "Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Factorial requested n ORIGINAL factorial factor4，不以小數近似替代。",
          "en": "Recompute Factorial requested n ORIGINAL factorial factor4 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "6",
          "4"
        ],
        "output": "24",
        "label": "Factorial requested n ORIGINAL factorial factor4"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "Factorial requested n ORIGINAL factorial factor5 := mul(24, 5)",
      "after": "Factorial requested n ORIGINAL factorial factor5 = 120",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "使用原操作數與明示次序作精確整數／有理計算，不用浮點近似。輸入界依題族：不把衍生值重驗成raw輸入，也不給BASE題庫另加SAFE或n64界。",
        "en": "Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Factorial requested n ORIGINAL factorial factor5，不以小數近似替代。",
          "en": "Recompute Factorial requested n ORIGINAL factorial factor5 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "24",
          "5"
        ],
        "output": "120",
        "label": "Factorial requested n ORIGINAL factorial factor5"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "Factorial requested n ORIGINAL factorial factor6 := mul(120, 6)",
      "after": "Factorial requested n ORIGINAL factorial factor6 = 720",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "使用原操作數與明示次序作精確整數／有理計算，不用浮點近似。輸入界依題族：不把衍生值重驗成raw輸入，也不給BASE題庫另加SAFE或n64界。",
        "en": "Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Factorial requested n ORIGINAL factorial factor6，不以小數近似替代。",
          "en": "Recompute Factorial requested n ORIGINAL factorial factor6 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "120",
          "6"
        ],
        "output": "720",
        "label": "Factorial requested n ORIGINAL factorial factor6"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "Factorial requested n ORIGINAL factorial factor7 := mul(720, 7)",
      "after": "Factorial requested n ORIGINAL factorial factor7 = 5040",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "使用原操作數與明示次序作精確整數／有理計算，不用浮點近似。輸入界依題族：不把衍生值重驗成raw輸入，也不給BASE題庫另加SAFE或n64界。",
        "en": "Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Factorial requested n ORIGINAL factorial factor7，不以小數近似替代。",
          "en": "Recompute Factorial requested n ORIGINAL factorial factor7 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "720",
          "7"
        ],
        "output": "5040",
        "label": "Factorial requested n ORIGINAL factorial factor7"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "Factorial requested n ORIGINAL factorial factor8 := mul(5040, 8)",
      "after": "Factorial requested n ORIGINAL factorial factor8 = 40320",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "使用原操作數與明示次序作精確整數／有理計算，不用浮點近似。輸入界依題族：不把衍生值重驗成raw輸入，也不給BASE題庫另加SAFE或n64界。",
        "en": "Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Factorial requested n ORIGINAL factorial factor8，不以小數近似替代。",
          "en": "Recompute Factorial requested n ORIGINAL factorial factor8 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "5040",
          "8"
        ],
        "output": "40320",
        "label": "Factorial requested n ORIGINAL factorial factor8"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "Factorial requested n ORIGINAL factorial factor9 := mul(40320, 9)",
      "after": "Factorial requested n ORIGINAL factorial factor9 = 362880",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "使用原操作數與明示次序作精確整數／有理計算，不用浮點近似。輸入界依題族：不把衍生值重驗成raw輸入，也不給BASE題庫另加SAFE或n64界。",
        "en": "Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Factorial requested n ORIGINAL factorial factor9，不以小數近似替代。",
          "en": "Recompute Factorial requested n ORIGINAL factorial factor9 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "40320",
          "9"
        ],
        "output": "362880",
        "label": "Factorial requested n ORIGINAL factorial factor9"
      }
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "Factorial requested n ORIGINAL factorial factor10 := mul(362880, 10)",
      "after": "Factorial requested n ORIGINAL factorial factor10 = 3628800",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "使用原操作數與明示次序作精確整數／有理計算，不用浮點近似。輸入界依題族：不把衍生值重驗成raw輸入，也不給BASE題庫另加SAFE或n64界。",
        "en": "Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Factorial requested n ORIGINAL factorial factor10，不以小數近似替代。",
          "en": "Recompute Factorial requested n ORIGINAL factorial factor10 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "362880",
          "10"
        ],
        "output": "3628800",
        "label": "Factorial requested n ORIGINAL factorial factor10"
      }
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "source roles = [Factorial requested n, 10]",
      "after": "derived = [3628800]",
      "relation": "implies",
      "rule_id": "Factorial_COMPLETE_product_EMPTY0and1_are1_not_input_n64_cap",
      "conditions": [],
      "explanation": {
        "zh-Hant": "階乘由完整2..n相乘，n0及1都是空積1。衍生值用任意精度整數，不用Number或科學記號，也不把其他題族n64界搬來。",
        "en": "Factorial multiplies the COMPLETE2..n factors;at n0/1 the empty product is1.Use arbitrary-precision integers,not Number/scientific notation or another family n64 cap."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向代回原輸入、欄位角色、論域及本步條件；一致性不表示結論能唯一恢復前提。有限核對、獨立審查、形式證明與發布分開標記。",
          "en": "Reverse-check the ORIGINAL inputs,field roles,domain and premises.Consistency does not imply unique recovery of premises.Keep finite checking,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "source roles = [10, 3628800]",
      "after": "derived = [3628800]",
      "relation": "implies",
      "rule_id": "Factorial_GIVEN_nfactorial_STRING_verified_not_assumed",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先完整重算n!並比對給定decimal字串，不能把未驗證的提示當答案oracle。字串須恰等原值，保所有位數。",
        "en": "Recompute the WHOLE n! and compare the GIVEN decimal STRING first;do not treat an unchecked hint as an answer oracle.Require the exact string,retaining every digit."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向代回原輸入、欄位角色、論域及本步條件；一致性不表示結論能唯一恢復前提。有限核對、獨立審查、形式證明與發布分開標記。",
          "en": "Reverse-check the ORIGINAL inputs,field roles,domain and premises.Consistency does not imply unique recovery of premises.Keep finite checking,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "Factorial ORIGINAL successor index n plus1 := sum(10, 1)",
      "after": "Factorial ORIGINAL successor index n plus1 = 11",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "使用原操作數與明示次序作精確整數／有理計算，不用浮點近似。輸入界依題族：不把衍生值重驗成raw輸入，也不給BASE題庫另加SAFE或n64界。",
        "en": "Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Factorial ORIGINAL successor index n plus1，不以小數近似替代。",
          "en": "Recompute Factorial ORIGINAL successor index n plus1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "10",
          "1"
        ],
        "output": "11",
        "label": "Factorial ORIGINAL successor index n plus1"
      }
    },
    {
      "id": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "Factorial ORIGINAL successor factor times verified n factorial := mul(11, 3628800)",
      "after": "Factorial ORIGINAL successor factor times verified n factorial = 39916800",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "使用原操作數與明示次序作精確整數／有理計算，不用浮點近似。輸入界依題族：不把衍生值重驗成raw輸入，也不給BASE題庫另加SAFE或n64界。",
        "en": "Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Factorial ORIGINAL successor factor times verified n factorial，不以小數近似替代。",
          "en": "Recompute Factorial ORIGINAL successor factor times verified n factorial with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "11",
          "3628800"
        ],
        "output": "39916800",
        "label": "Factorial ORIGINAL successor factor times verified n factorial"
      }
    },
    {
      "id": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "source roles = [10, 3628800, 11]",
      "after": "derived = [39916800]",
      "relation": "implies",
      "rule_id": "Factorial_SUCCESSOR_recurrence_uses_verified_nfactorial",
      "conditions": [],
      "explanation": {
        "zh-Hant": "在已驗n!後，用(n+1)!=(n+1)n!精確乘一次。n0時1!=1，不誤乘成0或把n!和(n+1)!互換。",
        "en": "AFTER verifying n!,apply(n+1)!=(n+1)n! by one exact multiplication.At n0 this gives1!=1,not0;do not exchange n!and(n+1)!."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反向代回原輸入、欄位角色、論域及本步條件；一致性不表示結論能唯一恢復前提。有限核對、獨立審查、形式證明與發布分開標記。",
          "en": "Reverse-check the ORIGINAL inputs,field roles,domain and premises.Consistency does not imply unique recovery of premises.Keep finite checking,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "result": "39916800"
  },
  "conclusion": {
    "canonical_value": "39916800",
    "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-20260926-factorial-foundations",
      "batch_path": "public/data/questions/batches/batch-20260926-factorial-foundations/questions.jsonl",
      "batch_sha256": "3786928793507c14ce8ef42ef1bac6b7a71130d7f6294f1ef9fcd7d02263e7d5",
      "original_line": 36,
      "original_line_sha256": "fc7f1c4309288bd949ca2219cfac1e3fe192cc1d3f7e125cf229bcda0a378056",
      "question_compact_sha256": "a60143841fca9cc3da4a76e50fc60326270d608637c1a8920c1020a8270a1622",
      "english_row_compact_sha256": "3c7f744e98d995b879c72e4d9efe75ed13bdc8fbcdf49514cf8e0b471891a67a",
      "english_manifest_path": "public/data/i18n/editions/batch-20260926-factorial-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20260926-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": "59d384ca3d958b27165f4295ff96efb155e7b12d3fb70bcdd0ed68148513c05a",
    "original_candidate_sha256": "ffe9b0f42782ce068c6345ef6e21140d0ce5acac3f0221b3f7de4d238a647e84",
    "original_candidate_line_sha256": "ffe9b0f42782ce068c6345ef6e21140d0ce5acac3f0221b3f7de4d238a647e84"
  },
  "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-20261026-011-p63",
    "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": "963099ffc9f754069fdce6792949a85ed350472465eaceb3038fa8011cf672bd",
      "question_first_witness_sha256": "d7655324c5d11528e1af90b757384cebd3313e19dc5f6f5b9b8ea7624fb06f6f"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 15,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "fbd6bbf9360b6e7cfe2eba087929096f119a21dfde8eddb87904ad5e4602174f",
      "arithmetic_steps": 11
    },
    "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."
  }
}