{
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  "question_id": "q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0059",
  "pilot_number": "R0048-24659",
  "number": 24659,
  "family_id": "qf-mod44-ogf-geometric-closed-form",
  "title": {
    "zh-Hant": "第 24659 題：解析普通生成函數",
    "en": "Question 24659: The Geometric Analytic Closed Form Requires Absolute r Times x Less Than One"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0059",
    "batch_id": "batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-analytic-ordinary-generating-function"
    ],
    "title_zh": "解析普通生成函數",
    "stem_zh": "A(x)=sum_n>=0 a_n x^n的完整sequence law與明示analytic domain。操作=ogf-geometric-closed-form；資料={\"ratio\":[9,10],\"x\":[1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(a_n=(9/10)^n_for_every_n>=0, 1/2, 9/20, 20/11, 是, A(x)=1/(1-rx)_only_on_absolute_rx_less_than_one)",
      "explanation_zh": "Wave6第七層核對=(a_n=(9/10)^n_for_every_n>=0, 1/2, 9/20, 20/11, 是, A(x)=1/(1-rx)_only_on_absolute_rx_less_than_one)；依完整明示資料獨立重算；finite-support、geometric或二階recurrence law均完整明示；recurrence保留initial boundary terms；formal identity不自動是analytic equality；radius外代值、finite-prefix決定完整OGF與未證termwise differentiation均禁止。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 2,
      "skills": [
        "解析普通生成函數"
      ]
    },
    "variant": {
      "family_id": "qf-mod44-ogf-geometric-closed-form",
      "seed": 202611133059,
      "parameters": {
        "ratio": [
          9,
          10
        ],
        "x": [
          1,
          2
        ],
        "domain": "Analytic_ordinary_generating_function",
        "operation": "ogf-geometric-closed-form",
        "semantic_template": "qf-mod44-ogf-geometric-closed-form/{\"ratio\":[9,10],\"x\":[1,2]}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_finite_or_symbolic_model",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-15",
      "contract_version": "ocme-question-factory-v0.1"
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  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0059",
    "batch_id": "batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
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      "family_id": "qf-mod44-ogf-geometric-closed-form",
      "seed": 202611133059,
      "semantic_template": "qf-mod44-ogf-geometric-closed-form/{\"ratio\":[9,10],\"x\":[1,2]}"
    },
    "fields": {
      "title": "The Geometric Analytic Closed Form Requires Absolute r Times x Less Than One",
      "stem": "Use a_n=(9/10)^n for every n≥0 and evaluation x=1/2. Give (coefficient_law, x, ratio_times_x, analytic_value, domain_certificate, certificate).",
      "answer": {
        "kind": "expression",
        "value": "(a_n=(9/10)^n_for_every_n>=0, 1/2, 9/20, 20/11, Yes, A(x)=1/(1-rx)_only_on_absolute_rx_less_than_one)",
        "explanation": "Require |r*x|<1 before asserting A(x)=1/(1−r*x). A finite denominator alone does not justify analytic equality outside convergence; formal power-series algebra has a separate scope. The exact result, retaining the requested order, is (a_n=(9/10)^n_for_every_n>=0, 1/2, 9/20, 20/11, Yes, A(x)=1/(1-rx)_only_on_absolute_rx_less_than_one)."
      },
      "skills": [
        "The Geometric Analytic Closed Form Requires Absolute r Times x Less Than One"
      ],
      "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": "A(x)=sum_n>=0 a_n x^n的完整sequence law與明示analytic domain。操作=ogf-geometric-closed-form；資料={\"ratio\":[9,10],\"x\":[1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
        "en": "Use a_n=(9/10)^n for every n≥0 and evaluation x=1/2. Give (coefficient_law, x, ratio_times_x, analytic_value, domain_certificate, certificate)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source roles = [ogf-geometric-closed-form]",
      "after": "derived = [[coefficient_law, x, ratio_times_x, analytic_value, domain_certificate, certificate]]",
      "relation": "implies",
      "rule_id": "source_ordered_fields_without_JS",
      "conditions": [],
      "explanation": {
        "zh-Hant": "直接列來源完整欄序，不需JavaScript；原題、布林旗標、數學物件與非聲稱保持分層。",
        "en": "List the full source field order without JavaScript,keeping original data,Boolean flags,mathematical objects and nonclaims distinct."
      },
      "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": "OGF ORIGINAL ratio times x := mul(9/10, 1/2)",
      "after": "OGF ORIGINAL ratio times x = 9/20",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，分母先核非零，不用浮點近似。",
        "en": "Substitute original roles and established intermediates exactly,checking nonzero denominators without floats."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 OGF ORIGINAL ratio times x，不以小數近似替代。",
          "en": "Recompute OGF ORIGINAL ratio times x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "9/10",
          "1/2"
        ],
        "output": "9/20",
        "label": "OGF ORIGINAL ratio times x"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "OGF geometric denominator := sub(1, 9/20)",
      "after": "OGF geometric denominator = 11/20",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，分母先核非零，不用浮點近似。",
        "en": "Substitute original roles and established intermediates exactly,checking nonzero denominators without floats."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 OGF geometric denominator，不以小數近似替代。",
          "en": "Recompute OGF geometric denominator with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "1",
          "9/20"
        ],
        "output": "11/20",
        "label": "OGF geometric denominator"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "OGF geometric analytic value := div(1, 11/20)",
      "after": "OGF geometric analytic value = 20/11",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，分母先核非零，不用浮點近似。",
        "en": "Substitute original roles and established intermediates exactly,checking nonzero denominators without floats."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 OGF geometric analytic value，不以小數近似替代。",
          "en": "Recompute OGF geometric analytic value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "1",
          "11/20"
        ],
        "output": "20/11",
        "label": "OGF geometric analytic value"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "source roles = [9/10, 1/2, 9/20]",
      "after": "derived = [是, 20/11, for EVERY N,S_N=(1-z^(N+1))/(1-z);z^(N+1)->0]",
      "relation": "implies",
      "rule_id": "OGF_WHOLE_geometric_limit_with_analytic_domain",
      "conditions": [],
      "explanation": {
        "zh-Hant": "整條a_n=r^n已明示；|rx|<1使分母非零並令幾何尾趨0，才可把形式恆等式解讀為analytic equality。域外分母非零本身不夠。",
        "en": "The whole law a_n=r^n is declared.|rx|<1 gives a nonzero denominator and vanishing geometric tail,justifying analytic equality.A nonzero denominator outside convergence is insufficient."
      },
      "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": [
      "a_n=(9/10)^n_for_every_n>=0",
      "1/2",
      "9/20",
      "20/11",
      true,
      "A(x)=1/(1-rx)_only_on_absolute_rx_less_than_one"
    ]
  },
  "conclusion": {
    "canonical_value": "(a_n=(9/10)^n_for_every_n>=0, 1/2, 9/20, 20/11, 是, A(x)=1/(1-rx)_only_on_absolute_rx_less_than_one)",
    "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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