{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0098",
  "pilot_number": "R0048-24698",
  "number": 24698,
  "family_id": "qf-mod44-ogf-recurrence-boundary-terms",
  "title": {
    "zh-Hant": "第 24698 題：解析普通生成函數",
    "en": "Question 24698: The Full Recurrence Keeps Both Initial Boundary Terms in Its OGF"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0098",
    "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-recurrence-boundary-terms；資料={\"a0\":23,\"a1\":24,\"x\":[1,5]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(23, 24, [23, 24, 47, 71, 118, 189, 307, 496, 803, 1299], 1/5, 116/5, 19/25, 580/19, A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved)",
      "explanation_zh": "Wave6第七層核對=(23, 24, [23, 24, 47, 71, 118, 189, 307, 496, 803, 1299], 1/5, 116/5, 19/25, 580/19, A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved)；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": 3,
      "skills": [
        "解析普通生成函數"
      ]
    },
    "variant": {
      "family_id": "qf-mod44-ogf-recurrence-boundary-terms",
      "seed": 202611133098,
      "parameters": {
        "a0": 23,
        "a1": 24,
        "x": [
          1,
          5
        ],
        "domain": "Analytic_ordinary_generating_function",
        "operation": "ogf-recurrence-boundary-terms",
        "semantic_template": "qf-mod44-ogf-recurrence-boundary-terms/{\"a0\":[23,1],\"a1\":[24,1],\"x\":[1,5]}"
      }
    },
    "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"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0098",
    "batch_id": "batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "8b8d6b54b7d47b23730f58bd1ddd71c41c54dd703271b7e2a214b53f480b3056",
      "family_id": "qf-mod44-ogf-recurrence-boundary-terms",
      "seed": 202611133098,
      "semantic_template": "qf-mod44-ogf-recurrence-boundary-terms/{\"a0\":[23,1],\"a1\":[24,1],\"x\":[1,5]}"
    },
    "fields": {
      "title": "The Full Recurrence Keeps Both Initial Boundary Terms in Its OGF",
      "stem": "Use the whole recurrence a_n=a_(n−1)+a_(n−2) for every n≥2, initial a_0=23, a_1=24, and x=1/5 with |x|<1/2. Give (a0, a1, coefficient_preview, x, numerator, denominator, analytic_value, certificate).",
      "answer": {
        "kind": "expression",
        "value": "(23, 24, [23, 24, 47, 71, 118, 189, 307, 496, 803, 1299], 1/5, 116/5, 19/25, 580/19, A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved)",
        "explanation": "The ten displayed coefficients are a preview of this complete law, not its definition. Retain numerator a0+(a1−a0)*x and denominator1−x−x^2. The declared domain is sufficient for analytic convergence, not an asserted exact radius for every initial pair. No unproved termwise differentiation or radius-outside substitution is inferred. The exact result, retaining the requested order, is (23, 24, [23, 24, 47, 71, 118, 189, 307, 496, 803, 1299], 1/5, 116/5, 19/25, 580/19, A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved)."
      },
      "skills": [
        "The Full Recurrence Keeps Both Initial Boundary Terms in Its OGF"
      ],
      "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-recurrence-boundary-terms；資料={\"a0\":23,\"a1\":24,\"x\":[1,5]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
        "en": "Use the whole recurrence a_n=a_(n−1)+a_(n−2) for every n≥2, initial a_0=23, a_1=24, and x=1/5 with |x|<1/2. Give (a0, a1, coefficient_preview, x, numerator, denominator, analytic_value, certificate)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source roles = [ogf-recurrence-boundary-terms]",
      "after": "derived = [[a0, a1, coefficient_preview, x, numerator, denominator, analytic_value, 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 complete recurrence coefficient2 := sum(23, 24)",
      "after": "OGF complete recurrence coefficient2 = 47",
      "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 complete recurrence coefficient2，不以小數近似替代。",
          "en": "Recompute OGF complete recurrence coefficient2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "23",
          "24"
        ],
        "output": "47",
        "label": "OGF complete recurrence coefficient2"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "OGF complete recurrence coefficient3 := sum(24, 47)",
      "after": "OGF complete recurrence coefficient3 = 71",
      "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 complete recurrence coefficient3，不以小數近似替代。",
          "en": "Recompute OGF complete recurrence coefficient3 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "24",
          "47"
        ],
        "output": "71",
        "label": "OGF complete recurrence coefficient3"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "OGF complete recurrence coefficient4 := sum(47, 71)",
      "after": "OGF complete recurrence coefficient4 = 118",
      "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 complete recurrence coefficient4，不以小數近似替代。",
          "en": "Recompute OGF complete recurrence coefficient4 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "47",
          "71"
        ],
        "output": "118",
        "label": "OGF complete recurrence coefficient4"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "OGF complete recurrence coefficient5 := sum(71, 118)",
      "after": "OGF complete recurrence coefficient5 = 189",
      "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 complete recurrence coefficient5，不以小數近似替代。",
          "en": "Recompute OGF complete recurrence coefficient5 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "71",
          "118"
        ],
        "output": "189",
        "label": "OGF complete recurrence coefficient5"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "OGF complete recurrence coefficient6 := sum(118, 189)",
      "after": "OGF complete recurrence coefficient6 = 307",
      "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 complete recurrence coefficient6，不以小數近似替代。",
          "en": "Recompute OGF complete recurrence coefficient6 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "118",
          "189"
        ],
        "output": "307",
        "label": "OGF complete recurrence coefficient6"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "OGF complete recurrence coefficient7 := sum(189, 307)",
      "after": "OGF complete recurrence coefficient7 = 496",
      "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 complete recurrence coefficient7，不以小數近似替代。",
          "en": "Recompute OGF complete recurrence coefficient7 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "189",
          "307"
        ],
        "output": "496",
        "label": "OGF complete recurrence coefficient7"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "OGF complete recurrence coefficient8 := sum(307, 496)",
      "after": "OGF complete recurrence coefficient8 = 803",
      "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 complete recurrence coefficient8，不以小數近似替代。",
          "en": "Recompute OGF complete recurrence coefficient8 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "307",
          "496"
        ],
        "output": "803",
        "label": "OGF complete recurrence coefficient8"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "OGF complete recurrence coefficient9 := sum(496, 803)",
      "after": "OGF complete recurrence coefficient9 = 1299",
      "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 complete recurrence coefficient9，不以小數近似替代。",
          "en": "Recompute OGF complete recurrence coefficient9 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "496",
          "803"
        ],
        "output": "1299",
        "label": "OGF complete recurrence coefficient9"
      }
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "OGF initial a1 minus a0 boundary := sub(24, 23)",
      "after": "OGF initial a1 minus a0 boundary = 1",
      "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 initial a1 minus a0 boundary，不以小數近似替代。",
          "en": "Recompute OGF initial a1 minus a0 boundary with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "24",
          "23"
        ],
        "output": "1",
        "label": "OGF initial a1 minus a0 boundary"
      }
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "OGF initial boundary times x := mul(1, 1/5)",
      "after": "OGF initial boundary times x = 1/5",
      "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 initial boundary times x，不以小數近似替代。",
          "en": "Recompute OGF initial boundary times x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "1/5"
        ],
        "output": "1/5",
        "label": "OGF initial boundary times x"
      }
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "OGF numerator with BOTH initial terms := sum(23, 1/5)",
      "after": "OGF numerator with BOTH initial terms = 116/5",
      "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 numerator with BOTH initial terms，不以小數近似替代。",
          "en": "Recompute OGF numerator with BOTH initial terms with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "23",
          "1/5"
        ],
        "output": "116/5",
        "label": "OGF numerator with BOTH initial terms"
      }
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "(1/5)^2",
      "after": "OGF original x square = 1/25",
      "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": [
          "1/5",
          "2"
        ],
        "output": "1/25",
        "label": "OGF original x square"
      }
    },
    {
      "id": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "OGF denominator one minus x := sub(1, 1/5)",
      "after": "OGF denominator one minus x = 4/5",
      "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 denominator one minus x，不以小數近似替代。",
          "en": "Recompute OGF denominator one minus x with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "1",
          "1/5"
        ],
        "output": "4/5",
        "label": "OGF denominator one minus x"
      }
    },
    {
      "id": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "OGF denominator subtract x squared := sub(4/5, 1/25)",
      "after": "OGF denominator subtract x squared = 19/25",
      "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 denominator subtract x squared，不以小數近似替代。",
          "en": "Recompute OGF denominator subtract x squared with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "4/5",
          "1/25"
        ],
        "output": "19/25",
        "label": "OGF denominator subtract x squared"
      }
    },
    {
      "id": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "OGF recurrence analytic closed value := div(116/5, 19/25)",
      "after": "OGF recurrence analytic closed value = 580/19",
      "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 recurrence analytic closed value，不以小數近似替代。",
          "en": "Recompute OGF recurrence analytic closed value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "116/5",
          "19/25"
        ],
        "output": "580/19",
        "label": "OGF recurrence analytic closed value"
      }
    },
    {
      "id": "S017",
      "uses": [
        "G001",
        "S016"
      ],
      "before": "source roles = [23, 24, 1/5, [23, 24, 47, 71, 118, 189, 307, 496, 803, 1299]]",
      "after": "derived = [116/5, 19/25, 580/19, (1-x-x^2)A=a0+(a1-a0)x]",
      "relation": "implies",
      "rule_id": "OGF_ALL_n_recurrence_convergence_and_reindex_boundary_terms",
      "conditions": [],
      "explanation": {
        "zh-Hant": "十項只是完整recurrence的預覽。取C=max(|a0|,|a1|)，歸納|a_n|<=C*2^n；|x|<1/2給絕對收斂而可移項重排。n0留下a0、n1留下(a1−a0)x，n>=2全消。分母>=1−|x|−|x|²>1/4。這是充分domain，不宣稱每組初值的exact radius或termwise derivative。",
        "en": "The ten terms preview a whole recurrence.Take C=max(|a0|,|a1|);induction gives |a_n|<=C*2^n,so |x|<1/2 ensures absolute convergence and valid reindexing.Degree0 leaves a0,degree1 leaves(a1−a0)x,and all later coefficients cancel.The denominator exceeds1/4.This sufficient domain is not an exact radius or a termwise-derivative claim."
      },
      "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": [
      "23",
      "24",
      [
        "23",
        "24",
        "47",
        "71",
        "118",
        "189",
        "307",
        "496",
        "803",
        "1299"
      ],
      "1/5",
      "116/5",
      "19/25",
      "580/19",
      "A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved"
    ]
  },
  "conclusion": {
    "canonical_value": "(23, 24, [23, 24, 47, 71, 118, 189, 307, 496, 803, 1299], 1/5, 116/5, 19/25, 580/19, A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved)",
    "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-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations",
      "batch_path": "public/data/questions/batches/batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations/questions.jsonl",
      "batch_sha256": "528d8bb148daa9519f9f78e207714bd01d13b8fc7e44ae1d64c36fc87228ff44",
      "original_line": 98,
      "original_line_sha256": "b6f6605d1dc667fa224f7b4afedbb90c9924c922fd314bfdac27f0a8a6bf260a",
      "question_compact_sha256": "8b8d6b54b7d47b23730f58bd1ddd71c41c54dd703271b7e2a214b53f480b3056",
      "english_row_compact_sha256": "6867df82eedd963ed3ef94f94551e9645a4c2dcc2e603d35a02a3ce71e71e570",
      "english_manifest_path": "public/data/i18n/editions/batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-analytic-ordinary-generating-function": {
        "sha256": "bb78af651663a5b612aa987ab6c1ce997dd023e539de44fa7304c0b7363cb94c",
        "english_sha256": "a1f2435f6e237f8605cfd2cb349ded6e3b6d8d24b3bcfd2a84fb13e2862ff835",
        "original_path": "public/data/mko/analytic-ordinary-generating-function.json",
        "english_path": "public/data/i18n/editions/mko-analytic-ordinary-generating-function.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "7f1649723787c6d7cf1abe85636fd3740f2c3cc1784071db7e38069361c81fd4",
    "original_candidate_sha256": "5b2dc4a82806559c15e57217aa0e4f06bf6eec035f9d13175d7a9d8104e76975",
    "original_candidate_line_sha256": "5b2dc4a82806559c15e57217aa0e4f06bf6eec035f9d13175d7a9d8104e76975"
  },
  "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-20261026-009-p49",
    "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": "049a9e07e69fead88d93b34f0937b9470a06d71c1a8d097058fc83554529bdd0",
      "question_first_witness_sha256": "0c71ebb418f3de347d232d159fc25e93036f4ab9166bcfb5ee5b4be86d5c55b9"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 17,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "02fc877cf4eeea8933cf39894931ea21f44f321dc00e7b075fbdc768009deed7",
      "arithmetic_steps": 15
    },
    "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."
  }
}