{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261014-advance-20261112-r02-taylor-polynomial-0040",
  "pilot_number": "R0047-24015",
  "number": 24015,
  "family_id": "qf-mod43-taylor-coefficient-construction",
  "title": {
    "zh-Hant": "第 24015 題：Taylor多項式",
    "en": "Question 24015: Each Shifted-Basis Coefficient Is the Derivative Divided by Its Factorial"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261014-advance-20261112-r02-taylor-polynomial-0040",
    "batch_id": "batch-20261014-advance-20261112-r02-taylor-polynomial-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-taylor-polynomial"
    ],
    "title_zh": "Taylor多項式",
    "stem_zh": "固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-coefficient-construction；資料={\"derivatives\":[15,30,45,60,75],\"center\":2,\"order\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(2, 4, [15, 30, 45/2, 10, 25/8], [[0, 15], [1, 30], [2, 45/2], [3, 10], [4, 25/8]], coefficient_k_equals_derivative_k_over_k_factorial)",
      "explanation_zh": "Wave6第六層核對=(2, 4, [15, 30, 45/2, 10, 25/8], [[0, 15], [1, 30], [2, 45/2], [3, 10], [4, 25/8]], coefficient_k_equals_derivative_k_over_k_factorial)；依完整明示資料獨立重算；係數必須是f^(k)(a)/k!且基底是(x-a)^k，n=0保留；Taylor polynomial本身不給remainder bound、series convergence或analyticity。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 2,
      "skills": [
        "Taylor多項式"
      ]
    },
    "variant": {
      "family_id": "qf-mod43-taylor-coefficient-construction",
      "seed": 202611122040,
      "parameters": {
        "derivatives": [
          15,
          30,
          45,
          60,
          75
        ],
        "center": 2,
        "order": 4,
        "domain": "Exact_finite_Taylor_polynomial",
        "operation": "taylor-coefficient-construction",
        "semantic_template": "qf-mod43-taylor-coefficient-construction/{\"center\":2,\"derivatives\":[15,30,45,60,75],\"order\":4}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_finite_or_symbolic_model",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "候選須主端重算；不構成人工審定、一般定理的有限枚舉替代、未宣告瑕積分、未明示係數律、Euler criterion循環、有限word等價、Lean或Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-14",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261014-advance-20261112-r02-taylor-polynomial-0040",
    "batch_id": "batch-20261014-advance-20261112-r02-taylor-polynomial-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "8f7e4bb47abd8e84efde4af2a8438f60233047e818f012bd7d01fcdf5495d5d5",
      "family_id": "qf-mod43-taylor-coefficient-construction",
      "seed": 202611122040,
      "semantic_template": "qf-mod43-taylor-coefficient-construction/{\"center\":2,\"derivatives\":[15,30,45,60,75],\"order\":4}"
    },
    "fields": {
      "title": "Each Shifted-Basis Coefficient Is the Derivative Divided by Its Factorial",
      "stem": "Use the derivative values=[15, 30, 45, 60, 75] at the fixed center=2, through declared order=4≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, shifted_coefficients, indexed_coefficients, certificate).",
      "answer": {
        "kind": "expression",
        "value": "(2, 4, [15, 30, 45/2, 10, 25/8], [[0, 15], [1, 30], [2, 45/2], [3, 10], [4, 25/8]], coefficient_k_equals_derivative_k_over_k_factorial)",
        "explanation": "Give exact coefficients f^(k)(a)/k! in increasing k and indexed rows [k,coefficient]. The basis is (x−a)^k, not x^k at a different center. No floating approximation or unscaled derivative substitution is allowed. The exact result, retaining the requested order, is (2, 4, [15, 30, 45/2, 10, 25/8], [[0, 15], [1, 30], [2, 45/2], [3, 10], [4, 25/8]], coefficient_k_equals_derivative_k_over_k_factorial)."
      },
      "skills": [
        "Each Shifted-Basis Coefficient Is the Derivative Divided by Its Factorial"
      ],
      "verification_warning": "This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."
    }
  },
  "givens": [
    {
      "id": "G001",
      "text": {
        "zh-Hant": "固定center、nonnegative order與完整zero-through-n derivative jet。操作=taylor-coefficient-construction；資料={\"derivatives\":[15,30,45,60,75],\"center\":2,\"order\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
        "en": "Use the derivative values=[15, 30, 45, 60, 75] at the fixed center=2, through declared order=4≥0. Orders0 through n, including f(a), are all required. These supplied values form a jet assumption; a numeric list alone does not prove differentiability of an unspecified function. Give (center, order, shifted_coefficients, indexed_coefficients, certificate)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source roles = [taylor-coefficient-construction]",
      "after": "derived = [[center, order, shifted_coefficients, indexed_coefficients, certificate]]",
      "relation": "implies",
      "rule_id": "source_ordered_fields_without_JS",
      "conditions": [],
      "explanation": {
        "zh-Hant": "本頁列完整來源欄序，不需JavaScript；gradient boundary依原mode選6／8／7欄，不補欄也不由答案猜mode。",
        "en": "List the source field order without JavaScript.Gradient boundary uses its original mode to choose6,8 or7 fields;no padding or answer-derived mode."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "source roles = [2, 4, [15, 30, 45, 60, 75]]",
      "after": "derived = [[0, 1, 2, 3, 4], 5, 是]",
      "relation": "implies",
      "rule_id": "Taylor_assumed_complete_derivative_jet_not_differentiability_test",
      "conditions": [],
      "explanation": {
        "zh-Hant": "原題把這些數據指定為同一函數在同一a的各階導數；數字清單並不驗證未知函數可微。0階是f(a)。",
        "en": "The data are assumed derivatives of the same function at the same a.A numeric list does not verify differentiability of an unknown function.Order0 is f(a)."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "source roles = [0]",
      "after": "derived = [1]",
      "relation": "implies",
      "rule_id": "empty_factorial_is_multiplicative_unit",
      "conditions": [],
      "explanation": {
        "zh-Hant": "0!是空乘積1，不是0。",
        "en": "The empty product0! is1,not0."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "Taylor coefficient0 derivative over factorial := div(15, 1)",
      "after": "Taylor coefficient0 derivative over factorial = 15",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor coefficient0 derivative over factorial，不以小數近似替代。",
          "en": "Recompute Taylor coefficient0 derivative over factorial with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "15",
          "1"
        ],
        "output": "15",
        "label": "Taylor coefficient0 derivative over factorial"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "Taylor k1 factorial step1 := mul(1, 1)",
      "after": "Taylor k1 factorial step1 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor k1 factorial step1，不以小數近似替代。",
          "en": "Recompute Taylor k1 factorial step1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "1"
        ],
        "output": "1",
        "label": "Taylor k1 factorial step1"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "Taylor coefficient1 derivative over factorial := div(30, 1)",
      "after": "Taylor coefficient1 derivative over factorial = 30",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor coefficient1 derivative over factorial，不以小數近似替代。",
          "en": "Recompute Taylor coefficient1 derivative over factorial with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "30",
          "1"
        ],
        "output": "30",
        "label": "Taylor coefficient1 derivative over factorial"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "Taylor k2 factorial step1 := mul(1, 1)",
      "after": "Taylor k2 factorial step1 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor k2 factorial step1，不以小數近似替代。",
          "en": "Recompute Taylor k2 factorial step1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "1"
        ],
        "output": "1",
        "label": "Taylor k2 factorial step1"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "Taylor k2 factorial step2 := mul(1, 2)",
      "after": "Taylor k2 factorial step2 = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor k2 factorial step2，不以小數近似替代。",
          "en": "Recompute Taylor k2 factorial step2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "2"
        ],
        "output": "2",
        "label": "Taylor k2 factorial step2"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "Taylor coefficient2 derivative over factorial := div(45, 2)",
      "after": "Taylor coefficient2 derivative over factorial = 45/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor coefficient2 derivative over factorial，不以小數近似替代。",
          "en": "Recompute Taylor coefficient2 derivative over factorial with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "45",
          "2"
        ],
        "output": "45/2",
        "label": "Taylor coefficient2 derivative over factorial"
      }
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "Taylor k3 factorial step1 := mul(1, 1)",
      "after": "Taylor k3 factorial step1 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor k3 factorial step1，不以小數近似替代。",
          "en": "Recompute Taylor k3 factorial step1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "1"
        ],
        "output": "1",
        "label": "Taylor k3 factorial step1"
      }
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "Taylor k3 factorial step2 := mul(1, 2)",
      "after": "Taylor k3 factorial step2 = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor k3 factorial step2，不以小數近似替代。",
          "en": "Recompute Taylor k3 factorial step2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "2"
        ],
        "output": "2",
        "label": "Taylor k3 factorial step2"
      }
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "Taylor k3 factorial step3 := mul(2, 3)",
      "after": "Taylor k3 factorial step3 = 6",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor k3 factorial step3，不以小數近似替代。",
          "en": "Recompute Taylor k3 factorial step3 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "3"
        ],
        "output": "6",
        "label": "Taylor k3 factorial step3"
      }
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "Taylor coefficient3 derivative over factorial := div(60, 6)",
      "after": "Taylor coefficient3 derivative over factorial = 10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor coefficient3 derivative over factorial，不以小數近似替代。",
          "en": "Recompute Taylor coefficient3 derivative over factorial with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "60",
          "6"
        ],
        "output": "10",
        "label": "Taylor coefficient3 derivative over factorial"
      }
    },
    {
      "id": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "Taylor k4 factorial step1 := mul(1, 1)",
      "after": "Taylor k4 factorial step1 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor k4 factorial step1，不以小數近似替代。",
          "en": "Recompute Taylor k4 factorial step1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "1"
        ],
        "output": "1",
        "label": "Taylor k4 factorial step1"
      }
    },
    {
      "id": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "Taylor k4 factorial step2 := mul(1, 2)",
      "after": "Taylor k4 factorial step2 = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor k4 factorial step2，不以小數近似替代。",
          "en": "Recompute Taylor k4 factorial step2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "2"
        ],
        "output": "2",
        "label": "Taylor k4 factorial step2"
      }
    },
    {
      "id": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "Taylor k4 factorial step3 := mul(2, 3)",
      "after": "Taylor k4 factorial step3 = 6",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor k4 factorial step3，不以小數近似替代。",
          "en": "Recompute Taylor k4 factorial step3 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "3"
        ],
        "output": "6",
        "label": "Taylor k4 factorial step3"
      }
    },
    {
      "id": "S017",
      "uses": [
        "G001",
        "S016"
      ],
      "before": "Taylor k4 factorial step4 := mul(6, 4)",
      "after": "Taylor k4 factorial step4 = 24",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor k4 factorial step4，不以小數近似替代。",
          "en": "Recompute Taylor k4 factorial step4 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "6",
          "4"
        ],
        "output": "24",
        "label": "Taylor k4 factorial step4"
      }
    },
    {
      "id": "S018",
      "uses": [
        "G001",
        "S017"
      ],
      "before": "Taylor coefficient4 derivative over factorial := div(75, 24)",
      "after": "Taylor coefficient4 derivative over factorial = 25/8",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor coefficient4 derivative over factorial，不以小數近似替代。",
          "en": "Recompute Taylor coefficient4 derivative over factorial with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "75",
          "24"
        ],
        "output": "25/8",
        "label": "Taylor coefficient4 derivative over factorial"
      }
    },
    {
      "id": "S019",
      "uses": [
        "G001",
        "S018"
      ],
      "before": "source roles = [2, 4, [15, 30, 45/2, 10, 25/8]]",
      "after": "derived = [T_n(x)=sum c_k*(x-center)^k, T_n^(j)(center)=declared derivative[j] for EVERY0<=j<=n, no remainder or infinite-series certificate]",
      "relation": "implies",
      "rule_id": "Taylor_shifted_basis_matches_derivatives_ALL_orders",
      "conditions": [],
      "explanation": {
        "zh-Hant": "在中心逐j求導，k<j消失、k>j仍含正次(x-a)而為0，唯k=j留下j!c_j；這證完整jet匹配。n=0保常數f(a)，不自動給原函數等式或餘項界。",
        "en": "For each j,terms k<j differentiate to0 and k>j still contain a positive power at a;only k=j yields j!*c_j.This matches the full jet.Order0 retains f(a),without automatically identifying the original function or bounding a remainder."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "result": [
      "2",
      4,
      [
        "15",
        "30",
        "45/2",
        "10",
        "25/8"
      ],
      [
        [
          0,
          "15"
        ],
        [
          1,
          "30"
        ],
        [
          2,
          "45/2"
        ],
        [
          3,
          "10"
        ],
        [
          4,
          "25/8"
        ]
      ],
      "coefficient_k_equals_derivative_k_over_k_factorial"
    ]
  },
  "conclusion": {
    "canonical_value": "(2, 4, [15, 30, 45/2, 10, 25/8], [[0, 15], [1, 30], [2, 45/2], [3, 10], [4, 25/8]], coefficient_k_equals_derivative_k_over_k_factorial)",
    "explanation": {
      "zh-Hant": "依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。",
      "en": "The full substitutions and premise checks give the original answer; source identity,format and status are unchanged."
    }
  },
  "definition_note": {
    "zh-Hant": "沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。",
    "en": "Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate."
  },
  "source": {
    "input_commit": "0f7be61e2c22317a4075561dc822c87d881a50db",
    "question_binding": {
      "batch_id": "batch-20261014-advance-20261112-r02-taylor-polynomial-foundations",
      "batch_path": "public/data/questions/batches/batch-20261014-advance-20261112-r02-taylor-polynomial-foundations/questions.jsonl",
      "batch_sha256": "c67ac0dd673d664623d9707d94303285afad1472da9173be45b762e8e2059407",
      "original_line": 40,
      "original_line_sha256": "426737ff21e723caf107ab2292ac06373a1e604d9bdceb3e7d99105ad8caf85b",
      "question_compact_sha256": "8f7e4bb47abd8e84efde4af2a8438f60233047e818f012bd7d01fcdf5495d5d5",
      "english_row_compact_sha256": "de1011503e6ecc8698af9a71584e2fe94430eda23d82b48f8ee694c02758af9c",
      "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261112-r02-taylor-polynomial-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261112-r02-taylor-polynomial-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-taylor-polynomial": {
        "sha256": "ee8f291872fc7e5a97cdbe468fffe5f7e4e3857373415ca1a65e5ab2b61a53a1",
        "english_sha256": "5009d7ebab6a826e0e1ca6a349eadfed04ba911cc721a2b103d2e4e14229974f",
        "original_path": "public/data/mko/taylor-polynomial.json",
        "english_path": "public/data/i18n/editions/mko-taylor-polynomial.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "b3aad195dad743751e4c8a4a6ce44f091c3824da1e52c21e9266d4ccd1dc1b68",
    "original_candidate_sha256": "8cf37eb1f4b037197c43a8a08638634482d329e66213b099e454db202d7e82e6",
    "original_candidate_line_sha256": "8cf37eb1f4b037197c43a8a08638634482d329e66213b099e454db202d7e82e6"
  },
  "historical_source_state": {
    "review_state": {
      "derivation": "candidate_authored_and_locally_checked",
      "source_question_state": "candidate",
      "independent_AI_or_human_review": "NOT_PERFORMED",
      "formalization": "not_formalized",
      "Lean": "NOT_RUN"
    },
    "scope_note": {
      "zh-Hant": "來源綁定教學推導候選；精確核對不等於獨立AI／人工審查或一般形式證明。原題、答案、MKO、公式、AST與歷史不變。",
      "en": "Source-bound pedagogical derivation candidate.Exact checking is not independent AI/human review or general formal proof.Original questions,answers,MKOs,formulas,ASTs and history remain unchanged."
    },
    "label": "Frozen producer metadata; current per-question review is separate below."
  },
  "revision": {
    "id": "review-20261026-009-p48",
    "review_date": "2026-10-26",
    "owner_authorized_reader_publication": true,
    "scope": "One hundred sixty family prototypes and their mathematical/bilingual reasoning variants, stems, boundary statements and every distinct explanation shape were reviewed in a context independent of the producer; parameter substitutions, source bindings, full math models, both answer strings and every arithmetic step were independently reconstructed per question from the original parameters only, with negative controls. This grouped method does not claim a separate literal AI reading of every repeated sentence or an extra AI voter.",
    "review_receipts": {
      "semantic_review_sha256": "9b43b9823e9bd2478d6caa5b57ffed161c2c8956af86974e901d5662ad994386",
      "question_first_witness_sha256": "eb97596068e2d5648033bab2b8f6ca9b653bbe8cb77c0f8e1d16d0c40e46f50b"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 19,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "a7d3e22bab1a1e7f393c5e878ccba7fb197c55ddac8052f24dff14f9feac12b8",
      "arithmetic_steps": 15
    },
    "uses_answer_as_premise": false,
    "human_review": "NOT_PERFORMED",
    "formalization": "not_formalized",
    "Lean": "NOT_RUN",
    "Canonical_truth_promotion": false,
    "authority": "Owner-authorized reviewed reader publication; original question/candidate states remain historical and unchanged."
  }
}