{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-real-exponential-function",
  "semantic_family_id": "sf-real-exponential-function",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "實指數函數"
  },
  "summary": {
    "zh-Hant": "實指數函數以對每個實數絕對收斂的冪級數E(x)=sum_{n≥0}x^n/n!定義，並滿足E(x+y)=E(x)E(y)與E(x)>0。"
  },
  "statement": {
    "zh-Hant": "完整coefficient law是a_n(x)=x^n/n!。coefficient ratio 1/(n+1)趨0，因此radius為infinity且對每個實x絕對收斂。Addition law需對所有degree使用二項式係數合併Cauchy product，且重排責任由絕對收斂支撐；單一declared degree只核一個係數。E(0)=1；E(x)E(-x)=1，且E(x/2)E(-x/2)=1先給E(x/2)非零，再由E(x)=E(x/2)^2推出嚴格正。有限partial sum不是E(x)，也不由本節點自動推出導數、log、ODE唯一性、injectivity或非零有理點的exact rational value。"
  },
  "formula": {
    "tex": "E(x)=S(x)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mrow><mi>E</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mrow><mi>S</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></math>",
    "semantic_ast": {
      "type": "equation",
      "lhs": {
        "type": "function_call",
        "callee": {
          "type": "symbol",
          "name": "E",
          "glyph": "E"
        },
        "arguments": [
          {
            "type": "symbol",
            "name": "x",
            "glyph": "x"
          }
        ]
      },
      "rhs": {
        "type": "function_call",
        "callee": {
          "type": "symbol",
          "name": "S",
          "glyph": "S"
        },
        "arguments": [
          {
            "type": "symbol",
            "name": "x",
            "glyph": "x"
          }
        ]
      }
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "7365fcb02200d1e064be39b3e96a921b365d1745ca092837e600916d0e1b881a"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "partial-sum題以exact rational x與BigInt/rational factorial arithmetic計算。"
    },
    {
      "id": "a2",
      "zh-Hant": "addition-law certificate分開記錄binomial coefficient identity與absolute-convergence重排依據。"
    }
  ],
  "symbols": [
    {
      "token": "E",
      "role": "real_exponential_function",
      "role_zh": "實指數函數",
      "scope": "representative_formula"
    },
    {
      "token": "x",
      "role": "real_argument",
      "role_zh": "實數參數",
      "scope": "representative_formula"
    },
    {
      "token": "S",
      "role": "absolutely_convergent_power_series",
      "role_zh": "絕對收斂冪級數",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-infinite-series",
      "title_zh": "無窮級數",
      "reason_zh": "series定義。"
    },
    {
      "id": "mko-real-number",
      "title_zh": "實數",
      "reason_zh": "argument domain。"
    },
    {
      "id": "mko-function-mapping",
      "title_zh": "函數映射",
      "reason_zh": "全實數函數。"
    },
    {
      "id": "mko-sequence-limit",
      "title_zh": "數列極限",
      "reason_zh": "partial sums。"
    },
    {
      "id": "mko-power-series-radius-of-convergence",
      "title_zh": "冪級數收斂半徑",
      "reason_zh": "radius infinity。"
    },
    {
      "id": "mko-factorial",
      "title_zh": "階乘",
      "reason_zh": "n!係數。"
    },
    {
      "id": "mko-absolute-convergence-of-series",
      "title_zh": "級數的絕對收斂",
      "reason_zh": "Cauchy product與重排責任。"
    },
    {
      "id": "mko-binomial-theorem",
      "title_zh": "二項式定理",
      "reason_zh": "addition-law係數合併。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "series law與finite partial sum必須分開。",
      "全域收斂由factorial tail控制。",
      "addition、inverse與positivity使用完整series性質，不以數值近似取代。"
    ],
    "common_misconception_zh": "把有限partial sum當exact exp，或只驗幾組x,y便宣稱addition law。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "本輪不物化log、微分方程、導數唯一性或超越性。",
    "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": "加入標準常態分布與實指數的微分性質。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-real-exponential-function.md 整理；2026-10-15 脈衝 R02（使用者手動接續 2026-11-13 工作序列；實際日期仍為 2026-10-15），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-15"
  }
}