{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-discrete-exponential-growth",
  "semantic_family_id": "sf-discrete-exponential-growth",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "離散指數成長模型"
  },
  "summary": {
    "zh-Hant": "固定時間步與固定非負因子 r 的離散模型滿足 x_(n+1)=r x_n，因此 x_n=x_0 r^n。"
  },
  "statement": {
    "zh-Hant": "固定一個時間步與非負常數因子 r，離散指數模型由 x_(n+1)=r x_n 定義。給初值 x_0，反覆代入或歸納得到 x_n=x_0 r^n；這是指定模型的代數結果，不是資料自動符合模型的證明。"
  },
  "formula": {
    "tex": "x(n+1)=M(r,x(n))",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mrow><mi>x</mi><mo>(</mo><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mo>=</mo><mrow><mi>M</mi><mo>(</mo><mi>r</mi><mo>,</mo><mrow><mi>x</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": "x",
          "glyph": "x"
        },
        "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": "symbol",
            "name": "r",
            "glyph": "r"
          },
          {
            "type": "function_call",
            "callee": {
              "type": "symbol",
              "name": "x",
              "glyph": "x"
            },
            "arguments": [
              {
                "type": "symbol",
                "name": "n",
                "glyph": "n"
              }
            ]
          }
        ]
      }
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "8dad62246fee9f06b1f58ff1df84e084c4dd74983e680b1f114be5e53bcbdaee"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "時間步固定且 n 從 0 起算；r 在整段期間保持不變。"
    },
    {
      "id": "a2",
      "zh-Hant": "題庫採精確有理初值與非負有理因子；不把它當連續時間模型或生物族群實測預測。"
    }
  ],
  "symbols": [
    {
      "token": "x",
      "role": "modeled_quantity",
      "role_zh": "模型量",
      "scope": "representative_formula"
    },
    {
      "token": "n",
      "role": "time_step_index",
      "role_zh": "時間步索引",
      "scope": "representative_formula"
    },
    {
      "token": "r",
      "role": "fixed_factor",
      "role_zh": "固定每步因子",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-sequence",
      "title_zh": "數列",
      "reason_zh": "離散狀態依自然數時間排列。"
    },
    {
      "id": "mko-power-root",
      "title_zh": "冪與根式",
      "reason_zh": "閉式使用 r 的自然數次冪。"
    },
    {
      "id": "mko-arithmetic-operations",
      "title_zh": "四則運算",
      "reason_zh": "遞迴逐步乘以固定因子。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "r>1 表示每步倍增，r=1 表示因子不變，0≤r<1 表示每步縮放；初值為零時序列仍全為零。",
      "遞迴式與閉式的等價可以逐步重算並以自然數歸納證明。",
      "真實生物系統可能受資源、隨機性、時間變動與量測誤差影響；固定 r 模型本身不證適配性。"
    ],
    "common_misconception_zh": "看到幾期成長就假設固定因子永遠不變，或把離散因子直接當連續成長率。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "本輪只核對固定步長、固定非負因子的精確離散模型；不是經驗預測。",
    "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 logistic 模型、時間變動因子、模型校準與殘差。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-discrete-exponential-growth.md 整理；2026-10-13 脈衝 R06（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"
  }
}