{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-taylor-remainder-theorem",
  "semantic_family_id": "sf-taylor-remainder-theorem",
  "type": "theorem",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "Taylor餘項定理"
  },
  "summary": {
    "zh-Hant": "足夠高階可微函數的Taylor approximation error由Lagrange remainder與derivative bound控制。"
  },
  "statement": {
    "zh-Hant": "明示center a、point x、order n與完整polynomial coefficients，直接算Taylor polynomial與remainder。degree≤n時remainder為0；degree n+1時以constant (n+1)th derivative及strictly intermediate midpoint給exact Lagrange witness。exact boundary只接受M至少為derivative supremum的valid bound，或0≤M<D的undersized witness；負M與其他假mode拒絕。finite jet不推出一般remainder theorem、Taylor-series equality或analyticity。"
  },
  "formula": {
    "tex": "R(f,a,x,n)=r",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mrow><mi>R</mi><mo>(</mo><mi>f</mi><mo>,</mo><mi>a</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>n</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": "a",
            "glyph": "a"
          },
          {
            "type": "symbol",
            "name": "x",
            "glyph": "x"
          },
          {
            "type": "symbol",
            "name": "n",
            "glyph": "n"
          }
        ]
      },
      "rhs": {
        "type": "symbol",
        "name": "r",
        "glyph": "r"
      }
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "cf327d85cbe90422ca7be36704c6d5bcb5509d9563f73101bb8de94324014007"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "polynomial instances採exact rational arithmetic。"
    },
    {
      "id": "a2",
      "zh-Hant": "general theorem的interval-wide differentiability與derivative bound明示為責任。"
    }
  ],
  "symbols": [
    {
      "token": "R",
      "role": "taylor_remainder_operator",
      "role_zh": "Taylor餘項",
      "scope": "representative_formula"
    },
    {
      "token": "f",
      "role": "differentiable_function",
      "role_zh": "可微函數",
      "scope": "representative_formula"
    },
    {
      "token": "a",
      "role": "expansion_center",
      "role_zh": "展開中心",
      "scope": "representative_formula"
    },
    {
      "token": "x",
      "role": "evaluation_point",
      "role_zh": "評估點",
      "scope": "representative_formula"
    },
    {
      "token": "n",
      "role": "taylor_order",
      "role_zh": "Taylor階數",
      "scope": "representative_formula"
    },
    {
      "token": "r",
      "role": "exact_remainder",
      "role_zh": "精確餘項",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-taylor-polynomial",
      "title_zh": "Taylor多項式",
      "reason_zh": "finite jet。"
    },
    {
      "id": "mko-higher-order-derivative",
      "title_zh": "高階導數",
      "reason_zh": "interval-wide n+1 derivative。"
    },
    {
      "id": "mko-mean-value-theorem",
      "title_zh": "均值定理",
      "reason_zh": "Lagrange remainder背景。"
    },
    {
      "id": "mko-absolute-value",
      "title_zh": "絕對值",
      "reason_zh": "remainder bound。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "完整計算Taylor polynomial與remainder。",
      "zero與degree n+1 exact witness分開。",
      "exact derivative supremum與undersized bound分層。"
    ],
    "common_misconception_zh": "從中心finite jet或少數samples推出一般Taylor remainder或analyticity。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "polynomial exact cases只證指定模型。",
    "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analytic Taylor-series convergence。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-taylor-remainder-theorem.md 整理；2026-10-15 脈衝 R03（使用者手動接續 2026-11-16 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"
  }
}