{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-compound-interest",
  "semantic_family_id": "sf-compound-interest",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "複利模型"
  },
  "summary": {
    "zh-Hant": "固定每期利率 i 且無額外現金流時，餘額滿足 B_(n+1)=(1+i)B_n，故 B_n=B_0(1+i)^n。"
  },
  "statement": {
    "zh-Hant": "在期別、計息時點與每期利率 i 已固定，且期間內無存提款或費用時，複利餘額以 B_(n+1)=(1+i)B_n 更新。若累積因子 1+i>0，n 期後為 B_0(1+i)^n。"
  },
  "formula": {
    "tex": "B(n+1)=M(1+i,B(n))",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mrow><mi>B</mi><mo>(</mo><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mo>=</mo><mrow><mi>M</mi><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mi>i</mi></mrow><mo>,</mo><mrow><mi>B</mi><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mo>)</mo></mrow></mrow></math>",
    "semantic_ast": {
      "type": "equation",
      "lhs": {
        "type": "function_call",
        "callee": {
          "type": "symbol",
          "name": "B",
          "glyph": "B"
        },
        "arguments": [
          {
            "type": "addition",
            "operands": [
              {
                "type": "symbol",
                "name": "n",
                "glyph": "n"
              },
              {
                "type": "number",
                "value": 1
              }
            ]
          }
        ]
      },
      "rhs": {
        "type": "function_call",
        "callee": {
          "type": "symbol",
          "name": "M",
          "glyph": "M"
        },
        "arguments": [
          {
            "type": "addition",
            "operands": [
              {
                "type": "number",
                "value": 1
              },
              {
                "type": "symbol",
                "name": "i",
                "glyph": "i"
              }
            ]
          },
          {
            "type": "function_call",
            "callee": {
              "type": "symbol",
              "name": "B",
              "glyph": "B"
            },
            "arguments": [
              {
                "type": "symbol",
                "name": "n",
                "glyph": "n"
              }
            ]
          }
        ]
      }
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "262e25717e0e62ee0aa736916186079f8afac68db7cc653f55dae76bd15fd9bc"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "每期利率、複利頻率與期數單位一致，期間內利率固定且無現金流、稅費或四捨五入。"
    },
    {
      "id": "a2",
      "zh-Hant": "本金為非負精確有理數且 1+i>0；模型計算不是投資建議、報價或市場預測。"
    }
  ],
  "symbols": [
    {
      "token": "B",
      "role": "account_balance",
      "role_zh": "期末餘額",
      "scope": "representative_formula"
    },
    {
      "token": "n",
      "role": "period_index",
      "role_zh": "期數索引",
      "scope": "representative_formula"
    },
    {
      "token": "i",
      "role": "periodic_rate",
      "role_zh": "固定每期利率",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-sequence",
      "title_zh": "數列",
      "reason_zh": "各期餘額形成離散序列。"
    },
    {
      "id": "mko-power-root",
      "title_zh": "冪與根式",
      "reason_zh": "n 期累積使用因子冪。"
    },
    {
      "id": "mko-ratio-proportion",
      "title_zh": "比與比例",
      "reason_zh": "利率與累積因子以比例表示。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "每期先把目前餘額乘以同一累積因子；反覆 n 次得到冪次公式。",
      "利息金額是期末餘額減本金；若有定期存款、提款、費用或變動利率，就不是本條目的單一冪次模型。",
      "年利率、月利率與名目／有效利率不能混用；必須先固定期別與轉換慣例。"
    ],
    "common_misconception_zh": "把單利 B_0(1+ni) 與複利 B_0(1+i)^n 混同，或混用不同期別利率。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "本輪只核對固定期率、無現金流的精確模型；不是投資建議或市場預測。",
    "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": "建立現值折現、名目／有效利率、年金與現金流時點。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-compound-interest.md 整理；2026-10-13 脈衝 R07（使用者手動接續 2026-10-20 工作序列；實際日期仍為 2026-10-13），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-13"
  }
}