{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-power-series-radius-of-convergence",
  "semantic_family_id": "sf-power-series-radius-of-convergence",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "冪級數收斂半徑"
  },
  "summary": {
    "zh-Hant": "冪級數在中心c周圍具有內部絕對收斂、外部發散的半徑R，邊界點須逐案判定。"
  },
  "statement": {
    "zh-Hant": "對sum_{n≥0} a_n(x-c)^n，R可為0、有限正數或infinity。若|x-c|<R則絕對收斂；若|x-c|>R則發散；|x-c|=R時不能由R單獨判定。本輪geometric law固定a_n=q^n；endpoint law固定a_n=R^(-n)/(n+1)^p；prefix boundary的兩序列在0≤n≤N皆為0，之後分別為q_1^(n-N-1)與q_2^(n-N-1)。Ratio test只在相應極限存在時可用，不能當一般定義。有限coefficient prefix不足以決定一般radius，且不得以只支援bounded-real supremum的規則排除R=0或infinity。"
  },
  "formula": {
    "tex": "R(f,c)=r",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mrow><mi>R</mi><mo>(</mo><mi>f</mi><mo>,</mo><mi>c</mi><mo>)</mo></mrow><mo>=</mo><mi>r</mi></mrow></math>",
    "semantic_ast": {
      "type": "equation",
      "lhs": {
        "type": "function_call",
        "callee": {
          "type": "symbol",
          "name": "R",
          "glyph": "R"
        },
        "arguments": [
          {
            "type": "symbol",
            "name": "f",
            "glyph": "f"
          },
          {
            "type": "symbol",
            "name": "c",
            "glyph": "c"
          }
        ]
      },
      "rhs": {
        "type": "symbol",
        "name": "r",
        "glyph": "r"
      }
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "2a40586ceecd3d37919a2e82c60647d00f6d7858197d456a868775fd8fb4a07c"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "每個exact family明示對全部n有效的coefficient law：geometric q^n、endpoint R^(-n)/(n+1)^p、factorial extremes或zero-prefix後的geometric tails。"
    },
    {
      "id": "a2",
      "zh-Hant": "radius與point位置以exact rational或0/infinity標記處理；endpoint另作獨立classification。"
    }
  ],
  "symbols": [
    {
      "token": "R",
      "role": "radius_of_convergence",
      "role_zh": "收斂半徑",
      "scope": "representative_formula"
    },
    {
      "token": "f",
      "role": "power_series",
      "role_zh": "冪級數",
      "scope": "representative_formula"
    },
    {
      "token": "c",
      "role": "series_center",
      "role_zh": "展開中心",
      "scope": "representative_formula"
    },
    {
      "token": "r",
      "role": "extended_nonnegative_radius",
      "role_zh": "擴充非負半徑",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-infinite-series",
      "title_zh": "無窮級數",
      "reason_zh": "pointwise級數收斂。"
    },
    {
      "id": "mko-supremum-and-infimum",
      "title_zh": "上確界與下確界",
      "reason_zh": "收斂集合半徑的極值描述。"
    },
    {
      "id": "mko-absolute-value",
      "title_zh": "絕對值",
      "reason_zh": "距中心距離與絕對收斂。"
    },
    {
      "id": "mko-power-root",
      "title_zh": "冪與根式",
      "reason_zh": "coefficient growth與root behavior。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "geometric family的完整law是a_n=q^n，故R=1/|q|。",
      "endpoint family用a_n=R^(-n)/(n+1)^p展示兩端可有不同收斂型態。",
      "R=0、R=infinity及兩條共享完整有限prefix但具有不同geometric tails的序列均為定義邊界。"
    ],
    "common_misconception_zh": "把endpoints自動算入、把ratio limit不存在當R不存在，或由有限係數前綴唯一決定R。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "本輪只對明示coefficient laws給exact證書；一般series仍需limsup或其他完整論證。",
    "evidence_refs": [],
    "producers": [
      {
        "id": "lean-mathlib",
        "adapter": "formal_system",
        "config_path": "formal/lean/lakefile.toml",
        "status": "configured"
      }
    ]
  },
  "formalization": {
    "status": "not_formalized",
    "target_systems": [
      "Lean 4",
      "Mathlib"
    ],
    "next_obligation_zh": "加入實指數函數與解析普通生成函數。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-power-series-radius-of-convergence.md 整理；2026-10-14 脈衝 R05（使用者手動接續 2026-11-12 工作序列；實際日期仍為 2026-10-14），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-14"
  }
}