{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261014-advance-20261029-r06-iid-sequence-0069",
  "pilot_number": "R0027-14469",
  "number": 14469,
  "family_id": "qf-mod29-iid-finite-factorization",
  "title": {
    "zh-Hant": "第 14469 題：獨立同分布序列",
    "en": "Question 14469: Multiply Every Ordered Atom Factor in the Finite Word"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261014-advance-20261029-r06-iid-sequence-0069",
    "batch_id": "batch-20261014-advance-20261029-r06-iid-sequence-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-iid-sequence"
    ],
    "title_zh": "獨立同分布序列",
    "stem_zh": "IID同時要求完整相同marginal與每個有限子族獨立。操作=iid-finite-factorization；資料={\"p\":[19,29],\"word\":[0,1,0,0]}。請依 [word,atom_factors,product,pass] 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "([0, 1, 0, 0], [10/29, 19/29, 10/29, 10/29], 19000/707281, 是)",
      "explanation_zh": "Wave4第三層核對=([0, 1, 0, 0], [10/29, 19/29, 10/29, 10/29], 19000/707281, 是)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；無限序列使用symbolic product-law cylinder證書；相同期望pairwise或固定prefix不足且不推出LLN/CLT。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 3,
      "skills": [
        "獨立同分布序列"
      ]
    },
    "variant": {
      "family_id": "qf-mod29-iid-finite-factorization",
      "seed": 202610296069,
      "parameters": {
        "p": [
          19,
          29
        ],
        "word": [
          0,
          1,
          0,
          0
        ],
        "domain": "Symbolic_Bernoulli_product_law",
        "operation": "iid-finite-factorization",
        "semantic_template": "qf-mod29-iid-finite-factorization/{\"p\":[19,29],\"word\":[0,1,0,0]}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_finite_model",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "產生端候選須由主端重算；不構成人工審定、任意無限定理、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、Lean執行或形式Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-14",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261014-advance-20261029-r06-iid-sequence-0069",
    "batch_id": "batch-20261014-advance-20261029-r06-iid-sequence-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "73bb8fda78dcaf7d230beca5121d3754f06bdffad48546daf58ecee7371eabbd",
      "family_id": "qf-mod29-iid-finite-factorization",
      "seed": 202610296069,
      "semantic_template": "qf-mod29-iid-finite-factorization/{\"p\":[19,29],\"word\":[0,1,0,0]}"
    },
    "fields": {
      "title": "Multiply Every Ordered Atom Factor in the Finite Word",
      "stem": "Use the explicitly declared Bernoulli product-law model with exact p=19/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. Ordered binary word=[0, 1, 0, 0]. Give (word,atom_factors,product,pass?).",
      "answer": {
        "kind": "expression",
        "value": "([0, 1, 0, 0], [10/29, 19/29, 10/29, 10/29], 19000/707281, Yes)",
        "explanation": "Assign factor p to each 1 and 1−p to each 0 in original word order, then multiply exactly. The factorization comes from the same product-law premise; it is not a test of an unknown dependent law. The exact result, retaining the requested order, is ([0, 1, 0, 0], [10/29, 19/29, 10/29, 10/29], 19000/707281, Yes)."
      },
      "skills": [
        "Multiply Every Ordered Atom Factor in the Finite Word"
      ],
      "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": "IID同時要求完整相同marginal與每個有限子族獨立。操作=iid-finite-factorization；資料={\"p\":[19,29],\"word\":[0,1,0,0]}。請依 [word,atom_factors,product,pass] 欄序回答。",
        "en": "Use the explicitly declared Bernoulli product-law model with exact p=19/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. Ordered binary word=[0, 1, 0, 0]. Give (word,atom_factors,product,pass?)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "Bernoulli zero-outcome mass := sub(1, 19/29)",
      "after": "Bernoulli zero-outcome mass = 10/29",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、完整配對與已核中間量精確計算；保留分母、正負號及零值。",
        "en": "Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Bernoulli zero-outcome mass，不以小數近似替代。",
          "en": "Recompute Bernoulli zero-outcome mass with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "1",
          "19/29"
        ],
        "output": "10/29",
        "label": "Bernoulli zero-outcome mass"
      }
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "raw Bernoulli parameter=[19, 29]",
      "after": "normalized p=19/29;mass at0=10/29;mass at1=19/29;symbolic product-law model explicitly declared",
      "relation": "implies",
      "rule_id": "Bernoulli_complete_marginal_and_product_law_premise",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留完整{0,1}邊際與宣告的乘積法則前提；不是從一組數據、相同期望或pairwise檢查猜無限IID。",
        "en": "Retain the complete{0,1} marginal and the explicitly declared product-law premise.Infinite IID is not inferred from a data set,equal means or pairwise checks."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "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": "original ordered finite binary word=[0, 1, 0, 0];p=19/29;one-minus-p=10/29",
      "after": "cylinder atom factorization uses the same declared symbolic product law",
      "relation": "implies",
      "rule_id": "IID_ordered_word_and_given_model_not_unknown_law_test",
      "conditions": [],
      "explanation": {
        "zh-Hant": "原字保留順序，1用p、0用1−p；此分解是模型前提，不是測一個未知相依模型後推得獨立。空字與p=0或1都保留。",
        "en": "Preserve word order,use p for1 and1−p for0.Factorization is a model premise,not a test proving independence of an unknown dependent law.Retain empty words and p=0 or1 boundaries."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "word position=0;original bit=0",
      "after": "aligned atom factor=10/29",
      "relation": "implies",
      "rule_id": "IID_each_original_bit_factor",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原位置選正確邊際因子，即使因子0也不略過；沒有用條件機率除法。",
        "en": "Choose the correct marginal factor at every original position,retaining zero factors without any conditional-probability division."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "ordered factor prefix product0 := mul(1, 10/29)",
      "after": "ordered factor prefix product0 = 10/29",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、完整配對與已核中間量精確計算；保留分母、正負號及零值。",
        "en": "Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 ordered factor prefix product0，不以小數近似替代。",
          "en": "Recompute ordered factor prefix product0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "10/29"
        ],
        "output": "10/29",
        "label": "ordered factor prefix product0"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "word position=1;original bit=1",
      "after": "aligned atom factor=19/29",
      "relation": "implies",
      "rule_id": "IID_each_original_bit_factor",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原位置選正確邊際因子，即使因子0也不略過；沒有用條件機率除法。",
        "en": "Choose the correct marginal factor at every original position,retaining zero factors without any conditional-probability division."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "ordered factor prefix product1 := mul(10/29, 19/29)",
      "after": "ordered factor prefix product1 = 190/841",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、完整配對與已核中間量精確計算；保留分母、正負號及零值。",
        "en": "Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 ordered factor prefix product1，不以小數近似替代。",
          "en": "Recompute ordered factor prefix product1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "10/29",
          "19/29"
        ],
        "output": "190/841",
        "label": "ordered factor prefix product1"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "word position=2;original bit=0",
      "after": "aligned atom factor=10/29",
      "relation": "implies",
      "rule_id": "IID_each_original_bit_factor",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原位置選正確邊際因子，即使因子0也不略過；沒有用條件機率除法。",
        "en": "Choose the correct marginal factor at every original position,retaining zero factors without any conditional-probability division."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "ordered factor prefix product2 := mul(190/841, 10/29)",
      "after": "ordered factor prefix product2 = 1900/24389",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、完整配對與已核中間量精確計算；保留分母、正負號及零值。",
        "en": "Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 ordered factor prefix product2，不以小數近似替代。",
          "en": "Recompute ordered factor prefix product2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "190/841",
          "10/29"
        ],
        "output": "1900/24389",
        "label": "ordered factor prefix product2"
      }
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "word position=3;original bit=0",
      "after": "aligned atom factor=10/29",
      "relation": "implies",
      "rule_id": "IID_each_original_bit_factor",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原位置選正確邊際因子，即使因子0也不略過；沒有用條件機率除法。",
        "en": "Choose the correct marginal factor at every original position,retaining zero factors without any conditional-probability division."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "ordered factor prefix product3 := mul(1900/24389, 10/29)",
      "after": "ordered factor prefix product3 = 19000/707281",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色、完整配對與已核中間量精確計算；保留分母、正負號及零值。",
        "en": "Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 ordered factor prefix product3，不以小數近似替代。",
          "en": "Recompute ordered factor prefix product3 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1900/24389",
          "10/29"
        ],
        "output": "19000/707281",
        "label": "ordered factor prefix product3"
      }
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "ordered atom factors=[10/29, 19/29, 10/29, 10/29]",
      "after": "finite product=19000/707281;model factorization check=是",
      "relation": "implies",
      "rule_id": "IID_finite_factorization_scope_not_new_infinite_proof",
      "conditions": [],
      "explanation": {
        "zh-Hant": "所有因子原序連乘；空清單從單位元1起算。通過是給定乘積模型內此有限計算的核對，不是用有限樣本外推任意無限IID。",
        "en": "Multiply every factor in original order,starting an empty list at identity1.Pass checks this finite calculation within the declared product model;it does not extrapolate arbitrary infinite IID from finite samples."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "result": [
      [
        0,
        1,
        0,
        0
      ],
      [
        "10/29",
        "19/29",
        "10/29",
        "10/29"
      ],
      "19000/707281",
      true
    ]
  },
  "conclusion": {
    "canonical_value": "([0, 1, 0, 0], [10/29, 19/29, 10/29, 10/29], 19000/707281, 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": {
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    "mko_bindings": {
      "mko-iid-sequence": {
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    },
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    "original_candidate_sha256": "3a6c6f53c0aded0fc93684acc73fc3439f2f8e29183cf54700957eb214d17bfb",
    "original_candidate_line_sha256": "3a6c6f53c0aded0fc93684acc73fc3439f2f8e29183cf54700957eb214d17bfb"
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  "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",
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    },
    "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-20261026-007-p28",
    "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": "e49bf5eaa968c09ebac4ad538ce58251671ed922c957cee1471cfc7b68934bfe",
      "question_first_witness_sha256": "a7976057c26051fab92342fc95d48af224466c0e1e4ab6ae4c8811c02772e6da"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 12,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "ca91484fc379ca8e3afb1583e0e0e3d3476d58eea13dedfc3348856f62be3a44",
      "arithmetic_steps": 5
    },
    "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."
  }
}