{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261010-advance-20261014-r05-factorial-0047",
  "pilot_number": "R0012-06797",
  "number": 6797,
  "family_id": "qf-density14-factorial-window-product",
  "title": {
    "zh-Hant": "第 6797 題：階乘窗口的乘積與因子索引",
    "en": "Question 6797: Factorial Window Product and Empty-Window Eligibility"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261010-advance-20261014-r05-factorial-0047",
    "batch_id": "batch-20261010-advance-20261014-r05-factorial-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-factorial"
    ],
    "title_zh": "階乘窗口的乘積與因子索引",
    "stem_zh": "N從0起；BigInt精確階乘；0≤lo=19≤hi=21≤64。依序答 (hi!/lo!, lo+1至hi的因子索引列表, 是否空窗口)。",
    "answer": {
      "kind": "expression",
      "value": "(420, [20, 21], 否)",
      "explanation_zh": "階乘窗口乘積與因子列=(420, [20, 21], 否)。只乘窗口中的因子；空窗口的乘積為1，不把它當0。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": [
        "階乘窗口的乘積與因子索引"
      ]
    },
    "variant": {
      "family_id": "qf-density14-factorial-window-product",
      "seed": 202610145047,
      "parameters": {
        "lo": 19,
        "hi": 21,
        "domain": "finite_N_factorial_protocol",
        "operation": "density14_factorial_window_product",
        "semantic_template": "qf-density14-factorial-window-product/[19,21]"
      }
    },
    "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-0047",
    "batch_id": "batch-20261010-advance-20261014-r05-factorial-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "492147c1879938fb3e674e152bba917e830b365fbe8530b6570109a1199a08f8",
      "family_id": "qf-density14-factorial-window-product",
      "seed": 202610145047,
      "semantic_template": "qf-density14-factorial-window-product/[19,21]"
    },
    "fields": {
      "title": "Factorial Window Product and Empty-Window Eligibility",
      "stem": "N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. 0≤lo=19≤hi=21≤64. Give (hi!/lo!, factor indices lo+1 through hi, is the window empty?).",
      "answer": {
        "kind": "expression",
        "value": "(420, [20, 21], No)",
        "explanation": "Multiply only the factors in the window; an empty window has product one, not zero. The exact result, retaining the requested order, is (420, [20, 21], No)."
      },
      "skills": [
        "Factorial Window Product and Empty-Window Eligibility"
      ],
      "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；0≤lo=19≤hi=21≤64。依序答 (hi!/lo!, lo+1至hi的因子索引列表, 是否空窗口)。",
        "en": "N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. 0≤lo=19≤hi=21≤64. Give (hi!/lo!, factor indices lo+1 through hi, is the window empty?)."
      }
    }
  ],
  "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": "lo=19;hi=21;window factors=[20, 21]",
      "after": "hi!/lo! is the product of exactly these factors",
      "relation": "implies",
      "rule_id": "factorial_positive_prefix_cancellation",
      "conditions": [],
      "explanation": {
        "zh-Hant": "lo!是正整數，可約掉共有的前綴乘積；lo=hi時沒有剩餘因子，商為1。",
        "en": "The positive integer lo! permits cancellation of the common prefix;when lo=hi,no factors remain and the quotient is1."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "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": "window through factor20 := mul(1, 20)",
      "after": "window through factor20 = 20",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只乘lo+1至hi範圍內的因子，不重做或加入窗口外因子。",
        "en": "Multiply only factors from lo+1 through hi,without outside factors."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 window through factor20，不以小數近似替代。",
          "en": "Recompute window through factor20 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "20"
        ],
        "output": "20",
        "label": "window through factor20"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "window through factor21 := mul(20, 21)",
      "after": "window through factor21 = 420",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只乘lo+1至hi範圍內的因子，不重做或加入窗口外因子。",
        "en": "Multiply only factors from lo+1 through hi,without outside factors."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 window through factor21，不以小數近似替代。",
          "en": "Recompute window through factor21 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "20",
          "21"
        ],
        "output": "420",
        "label": "window through factor21"
      }
    }
  ],
  "math_model": {
    "result": [
      "420",
      [
        20,
        21
      ],
      false
    ]
  },
  "conclusion": {
    "canonical_value": "(420, [20, 21], 否)",
    "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": 47,
      "original_line_sha256": "11ab5bd7444c8d56625ce9d7993d4a1ef0c31726d073bc0e52e2e2c595159010",
      "question_compact_sha256": "492147c1879938fb3e674e152bba917e830b365fbe8530b6570109a1199a08f8",
      "english_row_compact_sha256": "f857efe0e067aab4f4004117995d5af02f2e89a9db52e2a4f14305979f755140",
      "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": "c5f31b7eb35209d41d4e244884f1d406a8a784f9f2cebf0cff3fcaf1e1238299",
    "original_candidate_line_sha256": "c5f31b7eb35209d41d4e244884f1d406a8a784f9f2cebf0cff3fcaf1e1238299"
  },
  "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": 4,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "de57835d975a12cd57e4529a6d8fe12d2db922af3aa2c150c69f84f1606c9be3",
      "arithmetic_steps": 2
    },
    "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."
  }
}