{
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  "question_id": "q-20261013-advance-20261020-r06-discrete-exponential-growth-0046",
  "pilot_number": "R0002-01771",
  "number": 1771,
  "family_id": "qf-app20-growth-nth-term",
  "title": {
    "zh-Hant": "第 1771 題：離散固定因子模型",
    "en": "Question 1771: The Exact Natural-Indexed Term Including Zero Factor and Time"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261013-advance-20261020-r06-discrete-exponential-growth-0046",
    "batch_id": "batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-discrete-exponential-growth"
    ],
    "title_zh": "離散固定因子模型",
    "stem_zh": "固定時間步與精確固定因子。操作=growth-nth-term；資料={\"initial\":[12,1],\"factor\":[1,4],\"n\":0}。請依 [initial,factor,n,term] 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(12, 1/4, 0, 12)",
      "explanation_zh": "應用基礎核對=(12, 1/4, 0, 12)；依完整明示資料重算；精確有限切片核對不是一般定理或經驗預測；不是連續時間模型；不是經驗預測。"
    },
    "difficulty": {
      "band": "upper_secondary",
      "level": 3,
      "skills": [
        "離散固定因子模型"
      ]
    },
    "variant": {
      "family_id": "qf-app20-growth-nth-term",
      "seed": 202610206046,
      "parameters": {
        "initial": [
          12,
          1
        ],
        "factor": [
          1,
          4
        ],
        "n": 0,
        "domain": "Exact_discrete_exponential_model",
        "operation": "growth-nth-term",
        "semantic_template": "qf-app20-growth-nth-term/{\"factor\":[1,4],\"initial\":[12,1],\"n\":0}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_finite_model",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般定理、經驗預測、Lean執行或形式Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-13",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261013-advance-20261020-r06-discrete-exponential-growth-0046",
    "batch_id": "batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "76007063a97e5cbc3e9f2f12dd66f84730c404842d6722b0d799c1b2e285f21d",
      "family_id": "qf-app20-growth-nth-term",
      "seed": 202610206046,
      "semantic_template": "qf-app20-growth-nth-term/{\"factor\":[1,4],\"initial\":[12,1],\"n\":0}"
    },
    "fields": {
      "title": "The Exact Natural-Indexed Term Including Zero Factor and Time",
      "stem": "Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. initial=12/1, factor=1/4, n=0≥0. Give (initial, factor, n, initial*factor^n). The zeroth power is the empty product 1, including factor=0.",
      "answer": {
        "kind": "expression",
        "value": "(12, 1/4, 0, 12)",
        "explanation": "Retain the initial value at n=0; a zero factor gives zero only after a positive number of steps. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (12, 1/4, 0, 12)."
      },
      "skills": [
        "The Exact Natural-Indexed Term Including Zero Factor and Time"
      ],
      "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": "固定時間步與精確固定因子。操作=growth-nth-term；資料={\"initial\":[12,1],\"factor\":[1,4],\"n\":0}。請依 [initial,factor,n,term] 欄序回答。",
        "en": "Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. initial=12/1, factor=1/4, n=0≥0. Give (initial, factor, n, initial*factor^n). The zeroth power is the empty product 1, including factor=0."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "factor q=1/4≥0",
      "after": "X_(k+1)=q·X_k",
      "relation": "implies",
      "rule_id": "fixed_factor_recurrence",
      "conditions": [],
      "explanation": {
        "zh-Hant": "固定同一因子；這是題目明示的離散模型，不從少量數值猜測一般規律。",
        "en": "Use the same factor in the declared discrete model,rather than infer a general law from a few values."
      },
      "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": "X₀=12; n=0",
      "after": "n=0 uses the initial value; an empty product equals1",
      "relation": "implies",
      "rule_id": "initial_and_empty_product",
      "conditions": [],
      "explanation": {
        "zh-Hant": "n表示轉移次數；n=0不先更新，也不需要除以q，因此q=0仍可處理。",
        "en": "n counts updates;n=0 performs none.No division by q is required,so q=0 remains admissible."
      },
      "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": "X_k=X₀q^k ⇒ X_(k+1)=qX_k=X₀q^(k+1)",
      "after": "X_n=X₀q^n for every nonnegative integer n",
      "relation": "implies",
      "rule_id": "recurrence_induction",
      "conditions": [],
      "explanation": {
        "zh-Hant": "初項成立，乘q即可從k推到k+1；這是歸納論證，不是只靠有限表猜公式。",
        "en": "The initial case holds and multiplication by q gives the induction step;this is not a guess from a finite table."
      },
      "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": "(1/4)^0",
      "after": "q^n = 1",
      "relation": "implies",
      "rule_id": "integer_power",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以非負整數次有限乘積精確計算，零次為多項式中的乘法單位元。",
        "en": "Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity."
      },
      "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": {
        "operation": "integer_power",
        "inputs": [
          "1/4",
          "0"
        ],
        "output": "1",
        "label": "q^n"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "closed-form X_n := mul(12, 1)",
      "after": "closed-form X_n = 12",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "再用冪公式獨立代值，應與逐步更新相同。",
        "en": "Substitute into the power formula as a separate calculation;it must equal the stepwise update."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 closed-form X_n，不以小數近似替代。",
          "en": "Recompute closed-form X_n with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "12",
          "1"
        ],
        "output": "12",
        "label": "closed-form X_n"
      }
    }
  ],
  "math_model": {
    "result": [
      "12",
      "1/4",
      0,
      "12"
    ],
    "recurrence_values": [
      "12"
    ]
  },
  "conclusion": {
    "canonical_value": "(12, 1/4, 0, 12)",
    "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."
  },
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      "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."
    },
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}