{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-taylor-polynomial",
  "semantic_family_id": "sf-taylor-polynomial",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "Taylor多項式"
  },
  "summary": {
    "zh-Hant": "在中心a由f的零階到n階導數值依factorial縮放組成n階Taylor多項式。"
  },
  "statement": {
    "zh-Hant": "固定center a及order n≥0，並要求f^(k)(a)對0≤k≤n全部存在。T_n(x)=sum_{k=0}^n f^(k)(a)(x-a)^k/k!；n=0時為常數f(a)。Taylor polynomial本身不含remainder bound，也不保證無窮Taylor series收斂或等於f。若f本身是degree≤n的多項式則可exact重建；一般smooth函數不得由此外推analytic。"
  },
  "formula": {
    "tex": "T(f,a,n)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>T</mi><mo>(</mo><mi>f</mi><mo>,</mo><mi>a</mi><mo>,</mo><mi>n</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "T",
        "glyph": "T"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "f",
          "glyph": "f"
        },
        {
          "type": "symbol",
          "name": "a",
          "glyph": "a"
        },
        {
          "type": "symbol",
          "name": "n",
          "glyph": "n"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "6348a845feae270081d3a0f93b7e0894ad52c6e70c322602154659b0e0017f57"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "正向題族以完整derivative jet或以中心冪基底給出的精確有理／整數多項式係數。"
    },
    {
      "id": "a2",
      "zh-Hant": "所有factorial division使用exact rational arithmetic，不接受浮點近似或省略center shift。"
    }
  ],
  "symbols": [
    {
      "token": "T",
      "role": "taylor_polynomial_operator",
      "role_zh": "Taylor多項式算子",
      "scope": "representative_formula"
    },
    {
      "token": "f",
      "role": "function_with_finite_jet",
      "role_zh": "具有有限導數jet的函數",
      "scope": "representative_formula"
    },
    {
      "token": "a",
      "role": "expansion_center",
      "role_zh": "展開中心",
      "scope": "representative_formula"
    },
    {
      "token": "n",
      "role": "nonnegative_order",
      "role_zh": "非負階數",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-derivative-rules",
      "title_zh": "導數運算法則",
      "reason_zh": "導數值與多項式微分。"
    },
    {
      "id": "mko-polynomial",
      "title_zh": "多項式",
      "reason_zh": "有限Taylor和的代數物件。"
    },
    {
      "id": "mko-factorial",
      "title_zh": "階乘",
      "reason_zh": "k!係數縮放。"
    },
    {
      "id": "mko-higher-order-derivative",
      "title_zh": "高階導數",
      "reason_zh": "完整零至n階derivative jet。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "係數是f^(k)(a)/k!而不是裸導數。",
      "center shift必須使用(x-a)^k。",
      "finite polynomial不自帶remainder或analytic結論。"
    ],
    "common_misconception_zh": "漏掉factorial、在錯誤center使用derivative jet，或把Taylor polynomial直接等同原函數／無窮series。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "本輪只建立有限多項式；remainder、收斂與analyticity均須另證。",
    "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Taylor餘項定理。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-taylor-polynomial.md 整理；2026-10-14 脈衝 R02（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"
  }
}