{
  "schema_version": "ocme-derivation-reader-v1",
  "guide_id": "derivation-mko-analytic-ordinary-generating-function-01",
  "mko_id": "mko-analytic-ordinary-generating-function",
  "title": {
    "zh-Hant": "含初值邊界項的Fibonacci型OGF",
    "en": "A Fibonacci-type OGF with initial boundary terms"
  },
  "contract": {
    "schema_version": "ocme-derivation-guide-v0.1",
    "guide_id": "derivation-mko-analytic-ordinary-generating-function-01",
    "source": {
      "mko_id": "mko-analytic-ordinary-generating-function",
      "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
      "sha256": "bb78af651663a5b612aa987ab6c1ce997dd023e539de44fa7304c0b7363cb94c",
      "english_sha256": "a1f2435f6e237f8605cfd2cb349ded6e3b6d8d24b3bcfd2a84fb13e2862ff835"
    },
    "mode": "worked_example",
    "audience_prerequisites": [
      "work"
    ],
    "goal": {
      "zh-Hant": "在證明收斂的域內從遞迴推出OGF。",
      "en": "Derive the OGF from a recurrence on a justified convergence domain."
    },
    "givens": [
      {
        "id": "G1",
        "statement": {
          "zh-Hant": "a0=0、a1=1，所有n≥2有a_n=a_(n-1)+a_(n-2)；A(x)=sum_(n=0)^infinity a_n x^n。只在|x|<1/2推analytic等式。",
          "en": "a0=0,a1=1 and a_n=a_(n-1)+a_(n-2) for all n≥2; A(x)=sum_(n=0)^infinity a_n x^n. Derive analytic identities only on|x|<1/2."
        }
      }
    ],
    "steps": [
      {
        "id": "S01",
        "uses": [
          "G1",
          "work"
        ],
        "before_tex": "a_0=0\\le2^0,\\quad a_1=1\\le2^1",
        "after_tex": "a_{n-1}\\le2^{n-1},\\ a_{n-2}\\le2^{n-2}\\implies a_n\\le3\\cdot2^{n-2}\\le2^n",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "以初值為base、遞迴為step，歸納證0≤a_n≤2^n。",
          "en": "The initial values form the base and recurrence the step proving0≤a_n≤2^n by induction."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "work"
          ],
          "explanation": {
            "zh-Hant": "界涵蓋全部n，不從有限prefix猜radius。",
            "en": "The bound covers all n rather than guessing a radius from a prefix."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S02",
        "uses": [
          "G1",
          "S01",
          "work"
        ],
        "before_tex": "|x|<1/2",
        "after_tex": "\\sum_{n=0}^{\\infty}|a_nx^n|\\le\\sum_{n=0}^{\\infty}(2|x|)^n<\\infty",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "幾何級數比較證absolute convergence，允許後面shift與有限相加。",
          "en": "Geometric comparison proves absolute convergence, permitting later shifts and finite sums."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S01",
            "work"
          ],
          "explanation": {
            "zh-Hant": "2|x|<1是明示domain責任。",
            "en": "The domain obligation2|x|<1 is explicit."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S03",
        "uses": [
          "G1",
          "S02",
          "work"
        ],
        "before_tex": "A(x)=a_0+a_1x+\\sum_{n=2}^{\\infty}a_nx^n",
        "after_tex": "A(x)=x+\\sum_{n=2}^{\\infty}a_{n-1}x^n+\\sum_{n=2}^{\\infty}a_{n-2}x^n",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "完整保留a0與a1x邊界後才代遞迴。",
          "en": "Retain a0 and a1x boundary terms before substituting the recurrence."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S02",
            "work"
          ],
          "explanation": {
            "zh-Hant": "不能直接把A寫成xA+x²A而漏x。",
            "en": "Do not write A=xA+x²A and omit x."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S04",
        "uses": [
          "G1",
          "S03",
          "work"
        ],
        "before_tex": "\\sum_{n=2}^{\\infty}a_{n-1}x^n",
        "after_tex": "x\\sum_{j=1}^{\\infty}a_jx^j=x(A-a_0)=xA",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "j=n-1使下限為1，缺的a0明列後代0。",
          "en": "With j=n-1 the lower limit is 1; explicitly restore the missing a0 before substituting0."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S03",
            "work"
          ],
          "explanation": {
            "zh-Hant": "shift先處理boundary再簡化。",
            "en": "Handle the boundary before simplifying the shift."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S05",
        "uses": [
          "G1",
          "S04",
          "work"
        ],
        "before_tex": "\\sum_{n=2}^{\\infty}a_{n-2}x^n",
        "after_tex": "x^2\\sum_{j=0}^{\\infty}a_jx^j=x^2A",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "第二shift下限0，正好是完整A。",
          "en": "The second shift starts at 0 and gives the entire A."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S04",
            "work"
          ],
          "explanation": {
            "zh-Hant": "兩個shift的下限不同。",
            "en": "The two shifts have different lower limits."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S06",
        "uses": [
          "G1",
          "S05",
          "work"
        ],
        "before_tex": "A=x+xA+x^2A",
        "after_tex": "(1-x-x^2)A=x",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "移項並把A因子提出。",
          "en": "Move terms and factor A."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S05",
            "work"
          ],
          "explanation": {
            "zh-Hant": "方程在已證收斂域內成立。",
            "en": "The equation holds on the justified convergence domain."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S07",
        "uses": [
          "G1",
          "S06",
          "work"
        ],
        "before_tex": "|x+x^2|\\le|x|+|x|^2<3/4<1",
        "after_tex": "1-x-x^2\\ne0,\\quad A(x)=\\frac{x}{1-x-x^2}",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "先證分母非零，再合法相除。",
          "en": "Prove the denominator nonzero before division."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S06",
            "work"
          ],
          "explanation": {
            "zh-Hant": "非零分母本身不代表原級數在域外收斂。",
            "en": "A nonzero denominator alone does not give convergence outside the domain."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S08",
        "uses": [
          "G1",
          "S07",
          "work"
        ],
        "before_tex": "x=1/4",
        "after_tex": "A(1/4)=\\frac{1/4}{1-1/4-1/16}=4/11",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "1/4位於|x|<1/2，exact代值合法。",
          "en": "The point 1/4 lies in|x|<1/2, so exact evaluation is justified."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S07",
            "work"
          ],
          "explanation": {
            "zh-Hant": "沒有以有限partial sum冒充完整A值。",
            "en": "No finite partial sum is substituted for the full A value."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S09",
        "uses": [
          "G1",
          "S08",
          "work"
        ],
        "before_tex": "b_n=2^n,\\quad B(x)=\\sum_{n=0}^{\\infty}(2x)^n",
        "after_tex": "B(x)=1/(1-2x)\\quad(|2x|<1)",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "另一geometric family明列analytic domain；x=1時原級數發散。",
          "en": "For the geometric family state the analytic domain explicitly; at x=1 the original series diverges."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S08",
            "work"
          ],
          "explanation": {
            "zh-Hant": "rational式在x=1給-1不等於發散級數的和。",
            "en": "The rational value -1 at x=1 is not a sum of the divergent series."
          }
        },
        "leaf_basis": "work"
      }
    ],
    "result": {
      "zh-Hant": "|x|<1/2時A=x/(1-x-x²)，A(1/4)=4/11；未宣稱這是最大收斂域。",
      "en": "For|x|<1/2, A=x/(1-x-x²) and A(1/4)=4/11; no maximal-domain claim is made."
    },
    "negative_controls": [
      {
        "claim": {
          "zh-Hant": "A=xA+x²A即可，無需初值。",
          "en": "A=xA+x²A needs no initial values."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "漏a1x=x，會錯得A=0。",
          "en": "It omits a1x=x and incorrectly yields A=0."
        }
      },
      {
        "claim": {
          "zh-Hant": "B(1)=-1是2^n級數和。",
          "en": "B(1)=-1 sums the2^n series."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "|2x|=2≥1，原級數項不趨零。",
          "en": "Here|2x|=2≥1 and the series terms do not tend to zero."
        }
      },
      {
        "claim": {
          "zh-Hant": "前五個係數決定全部OGF。",
          "en": "The first five coefficients determine the whole OGF."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "可在第六項改係數而不改前五項，故須全序列遞迴。",
          "en": "Changing a later coefficient preserves the first five, so an all-index recurrence is needed."
        }
      }
    ],
    "status": "candidate",
    "gaps": [],
    "verification": {
      "structure": "checked_local_script",
      "mathematics": "candidate_argument",
      "formalization": "not_formalized"
    },
    "rule_glossary": [
      {
        "id": "mko_definition",
        "kind": "cited",
        "statement": {
          "zh-Hant": "coefficient extraction [x^n]A=a_n是編碼責任。有限partial sum只使用明示prefix。對a_n=r^n，A(x)=1/(1-rx)僅在|rx|<1成立。對a_n=a_{n-1}+a_{n-2}且初值a_0,a_1明示，A(x)=(a_0+(a_1-a_0)x)/(1-x-x^2)，boundary terms不可遺漏。formal power-series identity不自動是analytic equality；不得在radius外代值，有限coefficient prefix不決定完整OGF或radius，名稱中的analytic也不授權未證的逐項微分。",
          "en": "Coefficient extraction [x^n]A=a_n is the encoding obligation. A finite partial sum uses only its declared prefix. For a_n=r^n, A(x)=1/(1−r*x) holds analytically only when |r*x|<1. For a_n=a_(n−1)+a_(n−2) with explicit a0,a1, the OGF is (a0+(a1−a0)*x)/(1−x−x^2); initial boundary terms must not be omitted. A formal power-series identity is not automatically an analytic equality. Do not evaluate outside convergence, determine a whole OGF/radius from an unknown finite prefix or infer unproved termwise differentiation from the word analytic."
        },
        "source": {
          "mko_id": "mko-analytic-ordinary-generating-function",
          "sha256": "bb78af651663a5b612aa987ab6c1ce997dd023e539de44fa7304c0b7363cb94c",
          "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f"
        }
      },
      {
        "id": "work",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "數學歸納、幾何級數收斂與比較、absolute convergent series的index shift及有限線性組合；不假設OGF答案或逐項微分。",
          "en": "Induction, geometric convergence and comparison, index shifts and finite linear combinations of absolutely convergent series; neither the OGF answer nor termwise differentiation is assumed."
        }
      }
    ],
    "revision_log": [],
    "numeric_checks": [
      {
        "id": "N01",
        "lhs": "(1/4)/(1-1/4-1/16)",
        "rhs": "4/11"
      },
      {
        "id": "N02",
        "lhs": "1/(1-2*(1/4))",
        "rhs": "2"
      }
    ]
  },
  "presentation": {
    "md_scope": {
      "zh-Hant": "ordinary analytic OGF；不把formal identity任意代值。",
      "en": "An ordinary analytic OGF, not unrestricted evaluation of a formal identity."
    },
    "md_symbols": [
      [
        "A(x)",
        "a_n的ordinary生成函數",
        "ordinary generating function of a_n"
      ],
      [
        "[x^n]A",
        "係數a_n",
        "coefficient a_n"
      ]
    ],
    "md_error_controls": [
      {
        "title": "boundary terms / Boundary terms",
        "zh": "兩次shift分別從j=1與j=0，保留a0差異。",
        "en": "The shifts start at j=1 and j=0; retain the a0 difference."
      },
      {
        "title": "analytic域 / Analytic domain",
        "zh": "先以 2^n 的幾何比較證明級數在 |x|<1/2 域內絕對收斂，再代入該域內的 1/4。",
        "en": "First use the 2^n geometric comparison to prove absolute convergence on |x|<1/2, then evaluate at the point 1/4 inside that domain."
      }
    ],
    "md_open_items": []
  },
  "revision": {
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    "bilingual_sync": "complete_source_bound",
    "formal_Lean": "NOT_RUN",
    "reader_review": "142_of_142_source_bound",
    "wording_review": "original_Lead_decisions_resolved",
    "glossary_reader_paraphrases": [],
    "independent_AI_concurrence": "SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW",
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    "scope": "Owner-adopted reader revision. Lead and bilingual review concern the recorded text and premises; no new universal or Lean proof, canonical promotion or historical-label rewrite."
  }
}