{
  "schema_version": "ocme-derivation-reader-v1",
  "guide_id": "derivation-mko-taylor-remainder-theorem-01",
  "mko_id": "mko-taylor-remainder-theorem",
  "title": {
    "zh-Hant": "三次多項式的精確 Lagrange 餘項",
    "en": "An exact Lagrange remainder for a cubic"
  },
  "contract": {
    "schema_version": "ocme-derivation-guide-v0.1",
    "guide_id": "derivation-mko-taylor-remainder-theorem-01",
    "source": {
      "mko_id": "mko-taylor-remainder-theorem",
      "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
      "sha256": "ee3585c22f12885c07e98aac923c9f9ee997c7ade4196f27b93655ce96498057",
      "english_sha256": "cee54894be892c0a0578aa77bb18bcc713fb0b5f6010d561fffe2552fa63f60f"
    },
    "mode": "worked_example",
    "audience_prerequisites": [
      "algebra",
      "polynomial",
      "taylor_def"
    ],
    "goal": {
      "zh-Hant": "以 f(x)=x^3、a=1、n=2、x=2 核精確餘項及有效導數界。",
      "en": "For f(x)=x^3, a=1, n=2, x=2, audit the exact remainder and a qualified derivative bound."
    },
    "givens": [
      {
        "id": "G1",
        "statement": {
          "zh-Hant": "f(x)=x^3，a=1，n=2，評估點 2，連接區間 [1,2]。",
          "en": "f(x)=x^3, a=1, n=2, evaluation point 2, connecting interval [1,2]."
        }
      }
    ],
    "steps": [
      {
        "id": "S01",
        "uses": [
          "G1"
        ],
        "before_tex": "f'=3x^2,\\quad f''=6x,\\quad f'''=6",
        "after_tex": "(f(1),f'(1),f''(1))=(1,3,6)",
        "relation": "implies",
        "rule": "polynomial",
        "conditions": [],
        "explanation": {
          "zh-Hant": "全部所需階在整區間存在，三階導數更是常數。",
          "en": "Every required order exists on the entire interval, and the third derivative is constant."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1"
          ],
          "explanation": {
            "zh-Hant": "中心 jet 與區間三階資料都由完整多項式給出。",
            "en": "The full polynomial supplies both the center jet and interval-wide third derivative."
          }
        },
        "leaf_basis": "polynomial"
      },
      {
        "id": "S02",
        "uses": [
          "G1",
          "S01",
          "taylor_def"
        ],
        "before_tex": "T_2(x)=\\sum_{k=0}^2\\frac{f^{(k)}(1)}{k!}(x-1)^k",
        "after_tex": "T_2(x)=1+3(x-1)+3(x-1)^2",
        "relation": "definition",
        "rule": "taylor_def",
        "conditions": [],
        "explanation": {
          "zh-Hant": "逐階除 factorial 並保留中心位移。",
          "en": "Divide by each factorial and retain the center shift."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S01",
            "taylor_def"
          ],
          "explanation": {
            "zh-Hant": "二階 6/2!=3。",
            "en": "The quadratic coefficient is 6/2!=3."
          }
        },
        "leaf_basis": "taylor_def"
      },
      {
        "id": "S03",
        "uses": [
          "S02"
        ],
        "before_tex": "f(2)=8,\\quad T_2(2)=7",
        "after_tex": "R_2(2)=f(2)-T_2(2)=1",
        "relation": "equivalent",
        "rule": "algebra",
        "conditions": [],
        "explanation": {
          "zh-Hant": "先獨立算函數和近似，再作指定 signed remainder。",
          "en": "Evaluate the function and polynomial independently before forming the signed remainder."
        },
        "reverse_audit": {
          "kind": "inverse_equivalence",
          "required_ids": [
            "S02"
          ],
          "explanation": {
            "zh-Hant": "此處 remainder 是 f-T，不是反號。",
            "en": "The remainder here is f-T, not its negative."
          }
        },
        "leaf_basis": "algebra"
      },
      {
        "id": "S04",
        "uses": [
          "G1",
          "S01",
          "S03"
        ],
        "before_tex": "\\xi=(1+2)/2=3/2\\in(1,2)",
        "after_tex": "\\frac{f'''(\\xi)}{3!}(2-1)^3=\\frac6{6}=1=R_2(2)",
        "relation": "implies",
        "rule": "algebra",
        "conditions": [],
        "explanation": {
          "zh-Hant": "常三階導數使任意內點都可，明示中點為嚴格內部 Lagrange witness。",
          "en": "The constant third derivative makes every interior point valid; explicitly use the strict midpoint as a Lagrange witness."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S01",
            "S03"
          ],
          "explanation": {
            "zh-Hant": "評估點不同中心，才有此嚴格中點可選。",
            "en": "A strict midpoint exists here because the evaluation point differs from the center."
          }
        },
        "leaf_basis": "algebra"
      },
      {
        "id": "S05",
        "uses": [
          "S01",
          "taylor_def"
        ],
        "before_tex": "D=\\sup_{t\\in[1,2]}|f'''(t)|",
        "after_tex": "D=6,\\quad M=6\\ge D",
        "relation": "definition",
        "rule": "taylor_def",
        "conditions": [],
        "explanation": {
          "zh-Hant": "完整區間導數恆為 6，所以 supremum 已精確算出。",
          "en": "The derivative is constantly six on the whole interval, so the supremum is exact."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S01",
            "taylor_def"
          ],
          "explanation": {
            "zh-Hant": "不是只在中心或若干 sample 找最大。",
            "en": "This is not a maximum from the center or a few samples."
          }
        },
        "leaf_basis": "taylor_def"
      },
      {
        "id": "S06",
        "uses": [
          "S03",
          "S04",
          "S05"
        ],
        "before_tex": "|R_2(2)|\\le\\frac M{3!}|2-1|^3",
        "after_tex": "1\\le1\\quad(M=6)",
        "relation": "implies",
        "rule": "algebra",
        "conditions": [],
        "explanation": {
          "zh-Hant": "有效導數界給剛好取等的餘項界。",
          "en": "The qualified derivative bound gives a remainder estimate attained with equality."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S03",
            "S04",
            "S05"
          ],
          "explanation": {
            "zh-Hant": "界前提 M≥D 獨立於最後數值比較。",
            "en": "The premise M≥D is separate from the final numerical comparison."
          }
        },
        "leaf_basis": "algebra"
      },
      {
        "id": "S07",
        "uses": [
          "S03",
          "S05"
        ],
        "before_tex": "M=3<D=6",
        "after_tex": "\\frac M{3!}|2-1|^3=1/2<|R_2(2)|=1",
        "relation": "implies",
        "rule": "algebra",
        "conditions": [],
        "explanation": {
          "zh-Hant": "非負但過小的 M 既非導數界，且此點真的違反誤差不等式。",
          "en": "The nonnegative undersized M is not a derivative bound and actually violates the error inequality at this point."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S03",
            "S05"
          ],
          "explanation": {
            "zh-Hant": "負 M 更不合 bound 定義，但不是本 undersized witness 的混入。",
            "en": "A negative M would also be invalid, but is not being mixed into this nonnegative undersized witness."
          }
        },
        "leaf_basis": "algebra"
      },
      {
        "id": "S08",
        "uses": [
          "S02",
          "S05"
        ],
        "before_tex": "x=a=1",
        "after_tex": "R_2(1)=0;\\quad M=0\\text{ gives }0\\le0\\text{ but is not a bound on }|f'''|",
        "relation": "implies",
        "rule": "algebra",
        "conditions": [],
        "explanation": {
          "zh-Hant": "中心點的零因子可偶然讓錯 M 的數值式成立，不修補導數界資格；degree≤n 的多項式則整個 remainder 才為零。",
          "en": "At the center a zero factor may accidentally satisfy the inequality with invalid M; this does not qualify the derivative bound. A degree≤n polynomial separately has identically zero remainder."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S02",
            "S05"
          ],
          "explanation": {
            "zh-Hant": "區分 bound 資格與偶然單點不等式。",
            "en": "Separate bound qualification from an accidentally true pointwise inequality."
          }
        },
        "leaf_basis": "algebra"
      }
    ],
    "result": {
      "zh-Hant": "餘項 1、內點 witness 3/2、有效界 M=6；M=3 在此例失敗。",
      "en": "The remainder is 1, an interior witness is 3/2, M=6 is valid, and M=3 fails here."
    },
    "negative_controls": [
      {
        "claim": {
          "zh-Hant": "有限二階 jet 自動給任意連接區間的三階導數界。",
          "en": "A finite second-order jet automatically supplies a third-derivative bound on any connecting interval."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "f_K(x)=1+3(x-1)+3(x-1)^2+K(x-1)^3 的中心二階 jet 都是 (1,3,6)，但三階導數 6K 可任意大。",
          "en": "Every f_K(x)=1+3*(x-1)+3*(x-1)^2+K*(x-1)^3 has the same order-two jet (1,3,6) at one, but third derivative 6K is arbitrarily large."
        }
      },
      {
        "claim": {
          "zh-Hant": "M=3 是有效三階界。",
          "en": "M=3 bounds the third derivative."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "三階導數絕對值恆為 6。",
          "en": "The third derivative has constant magnitude 6."
        }
      },
      {
        "claim": {
          "zh-Hant": "x=a 時誤差式成立就證 M 合格。",
          "en": "The error inequality at x=a proves M is qualified."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "零位移可掩蓋錯界，S08 明示。",
          "en": "A zero displacement can hide an invalid bound, as S08 shows."
        }
      }
    ],
    "status": "candidate",
    "gaps": [],
    "verification": {
      "structure": "checked_local_script",
      "mathematics": "candidate_argument",
      "formalization": "not_formalized"
    },
    "rule_glossary": [
      {
        "id": "mko_definition",
        "kind": "cited",
        "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。",
          "en": "Declare center a, point x, order n and the complete polynomial coefficients, then compute the Taylor polynomial and exact remainder directly. Degree≤n gives zero remainder. For degree n+1 the next derivative is constant, and a strictly intermediate midpoint supplies an exact Lagrange witness when x differs from a. Boundary families accept a qualified M at least the exact derivative supremum D or an undersized witness0≤M<D, rejecting negative M and unknown modes. A finite center jet does not prove the general remainder theorem, equality with a Taylor series or analyticity."
        },
        "source": {
          "mko_id": "mko-taylor-remainder-theorem",
          "sha256": "ee3585c22f12885c07e98aac923c9f9ee997c7ade4196f27b93655ce96498057",
          "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f"
        }
      },
      {
        "id": "algebra",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "實數域的等式運算；除法前分母必非零，不等式乘正數保持方向。",
          "en": "Use real-field identities; division requires a nonzero denominator and multiplication by a positive number preserves inequalities."
        }
      },
      {
        "id": "polynomial",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "多項式在實數上連續且可逐項微分；d(x^k)/dx=k x^(k-1)，常數導數為零。",
          "en": "Real polynomials are continuous and differentiate termwise: the derivative of x^k is k*x^(k-1), and constants differentiate to zero."
        }
      },
      {
        "id": "taylor_def",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "中心 a 的 T_n 為 k=0 至 n 的 f^(k)(a)*(x-a)^k/k! 之和；remainder 為 f-T_n；導數 sup 取完整連接區間的絕對值最小上界。",
          "en": "T_n at center a sums f^(k)(a)*(x-a)^k/k! from zero through n; the remainder is f-T_n; a derivative supremum is its magnitude's least upper bound on the full connecting interval."
        }
      }
    ],
    "revision_log": [],
    "numeric_checks": [
      {
        "id": "remainder",
        "lhs": "2**3-(1+3+3)",
        "rhs": "1",
        "step": "S03"
      },
      {
        "id": "valid-bound",
        "lhs": "6/factorial(3)",
        "rhs": "1",
        "step": "S06"
      },
      {
        "id": "bad-bound",
        "lhs": "3/factorial(3)",
        "rhs": "1",
        "step": "S07",
        "relation": "<"
      }
    ]
  },
  "presentation": {
    "md_scope": {
      "zh-Hant": "完整 degree n+1 多項式；不以中心 finite jet 證一般區間定理。",
      "en": "A complete degree-n+1 polynomial, not a proof of a general interval theorem from a finite center jet."
    },
    "md_symbols": [
      [
        "R_2",
        "f-T 的餘項",
        "remainder f-T"
      ],
      [
        "D,M",
        "精確導數 sup 與候選界",
        "exact derivative supremum and candidate bound"
      ]
    ],
    "md_error_controls": [
      {
        "title": "區間責任 / Interval obligation",
        "zh": "M 要控制整個連接區間的所需導數。",
        "en": "M must control the required derivative on the entire connecting interval."
      },
      {
        "title": "符號及 factorial / Sign and factorial",
        "zh": "固定 R=f-T 並除 (n+1)!。",
        "en": "Keep R=f-T and divide by (n+1)!."
      }
    ],
    "md_open_items": []
  },
  "revision": {
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    "collection": "coverage-20261021",
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    "original_source_sha256": "dba2c7065fa51f643663241941c7b9b8831b346d4847714aa8db67441be9746d",
    "owner_adopted": true,
    "bilingual_sync": "complete_source_bound",
    "formal_Lean": "NOT_RUN",
    "reader_review": "142_of_142_source_bound",
    "wording_review": "original_Lead_decisions_resolved",
    "glossary_reader_paraphrases": [],
    "independent_AI_concurrence": "SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW",
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    "scope": "Owner-adopted reader revision. Lead and bilingual review concern the recorded text and premises; no new universal or Lean proof, canonical promotion or historical-label rewrite."
  }
}