{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261014-advance-20261112-r02-taylor-polynomial-0052",
  "pilot_number": "R0047-24052",
  "number": 24052,
  "family_id": "qf-mod43-taylor-exact-polynomial-reconstruction",
  "title": {
    "zh-Hant": "第 24052 題：Taylor多項式",
    "en": "Question 24052: A Declared Shifted Polynomial Reconstructs Exactly from Its Finite Jet"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261014-advance-20261112-r02-taylor-polynomial-0052",
    "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-exact-polynomial-reconstruction；資料={\"shifted_coefficients\":[2,4,6,8],\"center\":-11,\"point\":-12}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "([2, 4, 6, 8], [2, 4, 12, 48], [2, 4, 6, 8], -12, -4, -4, 是, degree_at_most_n_polynomial_is_exactly_reconstructed)",
      "explanation_zh": "Wave6第六層核對=([2, 4, 6, 8], [2, 4, 12, 48], [2, 4, 6, 8], -12, -4, -4, 是, degree_at_most_n_polynomial_is_exactly_reconstructed)；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": 1,
      "skills": [
        "Taylor多項式"
      ]
    },
    "variant": {
      "family_id": "qf-mod43-taylor-exact-polynomial-reconstruction",
      "seed": 202611122052,
      "parameters": {
        "shifted_coefficients": [
          2,
          4,
          6,
          8
        ],
        "center": -11,
        "point": -12,
        "domain": "Exact_finite_Taylor_polynomial",
        "operation": "taylor-exact-polynomial-reconstruction",
        "semantic_template": "qf-mod43-taylor-exact-polynomial-reconstruction/{\"center\":-11,\"point\":-12,\"shifted_coefficients\":[2,4,6,8]}"
      }
    },
    "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-0052",
    "batch_id": "batch-20261014-advance-20261112-r02-taylor-polynomial-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "bb6f775406de6afccb42a723fef254a47bb54c53e631a481107cd5e3b6ce29bb",
      "family_id": "qf-mod43-taylor-exact-polynomial-reconstruction",
      "seed": 202611122052,
      "semantic_template": "qf-mod43-taylor-exact-polynomial-reconstruction/{\"center\":-11,\"point\":-12,\"shifted_coefficients\":[2,4,6,8]}"
    },
    "fields": {
      "title": "A Declared Shifted Polynomial Reconstructs Exactly from Its Finite Jet",
      "stem": "Use shifted_coefficients=[2, 4, 6, 8] of the explicit polynomial sum c_k*(x−a)^k, center a=-11, evaluation point=-12. Give (shifted_coefficients, derivatives_at_center, reconstructed_coefficients, point, original_value, reconstructed_value, equal, certificate).",
      "answer": {
        "kind": "expression",
        "value": "([2, 4, 6, 8], [2, 4, 12, 48], [2, 4, 6, 8], -12, -4, -4, Yes, degree_at_most_n_polynomial_is_exactly_reconstructed)",
        "explanation": "For this polynomial, derivatives at a are k!*c_k, and dividing by those factorials recovers the same shifted coefficients. Compare exact original/reconstructed values. This finite polynomial equality is not an arbitrary smooth-function or infinite Taylor-series conclusion. The exact result, retaining the requested order, is ([2, 4, 6, 8], [2, 4, 12, 48], [2, 4, 6, 8], -12, -4, -4, Yes, degree_at_most_n_polynomial_is_exactly_reconstructed)."
      },
      "skills": [
        "A Declared Shifted Polynomial Reconstructs Exactly from Its Finite Jet"
      ],
      "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-exact-polynomial-reconstruction；資料={\"shifted_coefficients\":[2,4,6,8],\"center\":-11,\"point\":-12}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
        "en": "Use shifted_coefficients=[2, 4, 6, 8] of the explicit polynomial sum c_k*(x−a)^k, center a=-11, evaluation point=-12. Give (shifted_coefficients, derivatives_at_center, reconstructed_coefficients, point, original_value, reconstructed_value, equal, certificate)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source roles = [taylor-exact-polynomial-reconstruction]",
      "after": "derived = [[shifted_coefficients, derivatives_at_center, reconstructed_coefficients, point, original_value, reconstructed_value, equal, 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 = [-11, [2, 4, 6, 8]]",
      "after": "derived = [P(x)=sum c_k*(x-center)^k, 3]",
      "relation": "implies",
      "rule_id": "Taylor_finite_shifted_polynomial_whole_law",
      "conditions": [],
      "explanation": {
        "zh-Hant": "這裡整個有限多項式已明示；在a處第k導數僅k次項留下k!c_k，其他次項在代a時為0。不是任意光滑函數的無窮展開。",
        "en": "The whole finite polynomial is declared.At a,only its degree-k shifted monomial contributes k!*c_k to the kth derivative.This is not an infinite expansion of an arbitrary smooth function."
      },
      "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 derivative0 from shifted coefficient := mul(2, 1)",
      "after": "Taylor derivative0 from shifted coefficient = 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 derivative0 from shifted coefficient，不以小數近似替代。",
          "en": "Recompute Taylor derivative0 from shifted coefficient with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "1"
        ],
        "output": "2",
        "label": "Taylor derivative0 from shifted coefficient"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "Taylor recover coefficient0 := div(2, 1)",
      "after": "Taylor recover coefficient0 = 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 recover coefficient0，不以小數近似替代。",
          "en": "Recompute Taylor recover coefficient0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "2",
          "1"
        ],
        "output": "2",
        "label": "Taylor recover coefficient0"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "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": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "Taylor derivative1 from shifted coefficient := mul(4, 1)",
      "after": "Taylor derivative1 from shifted coefficient = 4",
      "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 derivative1 from shifted coefficient，不以小數近似替代。",
          "en": "Recompute Taylor derivative1 from shifted coefficient with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "4",
          "1"
        ],
        "output": "4",
        "label": "Taylor derivative1 from shifted coefficient"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "Taylor recover coefficient1 := div(4, 1)",
      "after": "Taylor recover coefficient1 = 4",
      "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 recover coefficient1，不以小數近似替代。",
          "en": "Recompute Taylor recover coefficient1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "4",
          "1"
        ],
        "output": "4",
        "label": "Taylor recover coefficient1"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "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": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "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": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "Taylor derivative2 from shifted coefficient := mul(6, 2)",
      "after": "Taylor derivative2 from shifted coefficient = 12",
      "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 derivative2 from shifted coefficient，不以小數近似替代。",
          "en": "Recompute Taylor derivative2 from shifted coefficient with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "6",
          "2"
        ],
        "output": "12",
        "label": "Taylor derivative2 from shifted coefficient"
      }
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "Taylor recover coefficient2 := div(12, 2)",
      "after": "Taylor recover coefficient2 = 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 recover coefficient2，不以小數近似替代。",
          "en": "Recompute Taylor recover coefficient2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "12",
          "2"
        ],
        "output": "6",
        "label": "Taylor recover coefficient2"
      }
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "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": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "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": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "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": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "Taylor derivative3 from shifted coefficient := mul(8, 6)",
      "after": "Taylor derivative3 from shifted coefficient = 48",
      "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 derivative3 from shifted coefficient，不以小數近似替代。",
          "en": "Recompute Taylor derivative3 from shifted coefficient with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "8",
          "6"
        ],
        "output": "48",
        "label": "Taylor derivative3 from shifted coefficient"
      }
    },
    {
      "id": "S017",
      "uses": [
        "G001",
        "S016"
      ],
      "before": "Taylor recover coefficient3 := div(48, 6)",
      "after": "Taylor recover coefficient3 = 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 recover coefficient3，不以小數近似替代。",
          "en": "Recompute Taylor recover coefficient3 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "48",
          "6"
        ],
        "output": "8",
        "label": "Taylor recover coefficient3"
      }
    },
    {
      "id": "S018",
      "uses": [
        "G001",
        "S017"
      ],
      "before": "Taylor SAME point minus declared center := sub(-12, -11)",
      "after": "Taylor SAME point minus declared center = -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 SAME point minus declared center，不以小數近似替代。",
          "en": "Recompute Taylor SAME point minus declared center with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "-12",
          "-11"
        ],
        "output": "-1",
        "label": "Taylor SAME point minus declared center"
      }
    },
    {
      "id": "S019",
      "uses": [
        "G001",
        "S018"
      ],
      "before": "Taylor original Horner product3 := mul(0, -1)",
      "after": "Taylor original Horner product3 = 0",
      "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 original Horner product3，不以小數近似替代。",
          "en": "Recompute Taylor original Horner product3 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "-1"
        ],
        "output": "0",
        "label": "Taylor original Horner product3"
      }
    },
    {
      "id": "S020",
      "uses": [
        "G001",
        "S019"
      ],
      "before": "Taylor original Horner sum3 := sum(0, 8)",
      "after": "Taylor original Horner sum3 = 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 original Horner sum3，不以小數近似替代。",
          "en": "Recompute Taylor original Horner sum3 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "8"
        ],
        "output": "8",
        "label": "Taylor original Horner sum3"
      }
    },
    {
      "id": "S021",
      "uses": [
        "G001",
        "S020"
      ],
      "before": "Taylor original Horner product2 := mul(8, -1)",
      "after": "Taylor original Horner product2 = -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 original Horner product2，不以小數近似替代。",
          "en": "Recompute Taylor original Horner product2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "8",
          "-1"
        ],
        "output": "-8",
        "label": "Taylor original Horner product2"
      }
    },
    {
      "id": "S022",
      "uses": [
        "G001",
        "S021"
      ],
      "before": "Taylor original Horner sum2 := sum(-8, 6)",
      "after": "Taylor original Horner sum2 = -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 original Horner sum2，不以小數近似替代。",
          "en": "Recompute Taylor original Horner sum2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-8",
          "6"
        ],
        "output": "-2",
        "label": "Taylor original Horner sum2"
      }
    },
    {
      "id": "S023",
      "uses": [
        "G001",
        "S022"
      ],
      "before": "Taylor original Horner product1 := mul(-2, -1)",
      "after": "Taylor original Horner product1 = 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 original Horner product1，不以小數近似替代。",
          "en": "Recompute Taylor original Horner product1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-2",
          "-1"
        ],
        "output": "2",
        "label": "Taylor original Horner product1"
      }
    },
    {
      "id": "S024",
      "uses": [
        "G001",
        "S023"
      ],
      "before": "Taylor original Horner sum1 := sum(2, 4)",
      "after": "Taylor original Horner sum1 = 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 original Horner sum1，不以小數近似替代。",
          "en": "Recompute Taylor original Horner sum1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "2",
          "4"
        ],
        "output": "6",
        "label": "Taylor original Horner sum1"
      }
    },
    {
      "id": "S025",
      "uses": [
        "G001",
        "S024"
      ],
      "before": "Taylor original Horner product0 := mul(6, -1)",
      "after": "Taylor original Horner product0 = -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 original Horner product0，不以小數近似替代。",
          "en": "Recompute Taylor original Horner product0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "6",
          "-1"
        ],
        "output": "-6",
        "label": "Taylor original Horner product0"
      }
    },
    {
      "id": "S026",
      "uses": [
        "G001",
        "S025"
      ],
      "before": "Taylor original Horner sum0 := sum(-6, 2)",
      "after": "Taylor original Horner sum0 = -4",
      "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 original Horner sum0，不以小數近似替代。",
          "en": "Recompute Taylor original Horner sum0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-6",
          "2"
        ],
        "output": "-4",
        "label": "Taylor original Horner sum0"
      }
    },
    {
      "id": "S027",
      "uses": [
        "G001",
        "S026"
      ],
      "before": "Taylor reconstructed Horner product3 := mul(0, -1)",
      "after": "Taylor reconstructed Horner product3 = 0",
      "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 reconstructed Horner product3，不以小數近似替代。",
          "en": "Recompute Taylor reconstructed Horner product3 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "-1"
        ],
        "output": "0",
        "label": "Taylor reconstructed Horner product3"
      }
    },
    {
      "id": "S028",
      "uses": [
        "G001",
        "S027"
      ],
      "before": "Taylor reconstructed Horner sum3 := sum(0, 8)",
      "after": "Taylor reconstructed Horner sum3 = 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 reconstructed Horner sum3，不以小數近似替代。",
          "en": "Recompute Taylor reconstructed Horner sum3 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "8"
        ],
        "output": "8",
        "label": "Taylor reconstructed Horner sum3"
      }
    },
    {
      "id": "S029",
      "uses": [
        "G001",
        "S028"
      ],
      "before": "Taylor reconstructed Horner product2 := mul(8, -1)",
      "after": "Taylor reconstructed Horner product2 = -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 reconstructed Horner product2，不以小數近似替代。",
          "en": "Recompute Taylor reconstructed Horner product2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "8",
          "-1"
        ],
        "output": "-8",
        "label": "Taylor reconstructed Horner product2"
      }
    },
    {
      "id": "S030",
      "uses": [
        "G001",
        "S029"
      ],
      "before": "Taylor reconstructed Horner sum2 := sum(-8, 6)",
      "after": "Taylor reconstructed Horner sum2 = -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 reconstructed Horner sum2，不以小數近似替代。",
          "en": "Recompute Taylor reconstructed Horner sum2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-8",
          "6"
        ],
        "output": "-2",
        "label": "Taylor reconstructed Horner sum2"
      }
    },
    {
      "id": "S031",
      "uses": [
        "G001",
        "S030"
      ],
      "before": "Taylor reconstructed Horner product1 := mul(-2, -1)",
      "after": "Taylor reconstructed Horner product1 = 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 reconstructed Horner product1，不以小數近似替代。",
          "en": "Recompute Taylor reconstructed Horner product1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-2",
          "-1"
        ],
        "output": "2",
        "label": "Taylor reconstructed Horner product1"
      }
    },
    {
      "id": "S032",
      "uses": [
        "G001",
        "S031"
      ],
      "before": "Taylor reconstructed Horner sum1 := sum(2, 4)",
      "after": "Taylor reconstructed Horner sum1 = 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 reconstructed Horner sum1，不以小數近似替代。",
          "en": "Recompute Taylor reconstructed Horner sum1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "2",
          "4"
        ],
        "output": "6",
        "label": "Taylor reconstructed Horner sum1"
      }
    },
    {
      "id": "S033",
      "uses": [
        "G001",
        "S032"
      ],
      "before": "Taylor reconstructed Horner product0 := mul(6, -1)",
      "after": "Taylor reconstructed Horner product0 = -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 reconstructed Horner product0，不以小數近似替代。",
          "en": "Recompute Taylor reconstructed Horner product0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "6",
          "-1"
        ],
        "output": "-6",
        "label": "Taylor reconstructed Horner product0"
      }
    },
    {
      "id": "S034",
      "uses": [
        "G001",
        "S033"
      ],
      "before": "Taylor reconstructed Horner sum0 := sum(-6, 2)",
      "after": "Taylor reconstructed Horner sum0 = -4",
      "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 reconstructed Horner sum0，不以小數近似替代。",
          "en": "Recompute Taylor reconstructed Horner sum0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-6",
          "2"
        ],
        "output": "-4",
        "label": "Taylor reconstructed Horner sum0"
      }
    },
    {
      "id": "S035",
      "uses": [
        "G001",
        "S034"
      ],
      "before": "source roles = [[2, 4, 6, 8], [2, 4, 6, 8], -11, -12]",
      "after": "derived = [是, -4, -4, 是]",
      "relation": "implies",
      "rule_id": "Taylor_ALL_coefficients_recovered_before_point_equality",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先逐係數除回k!得到同一整個多項式，再在同一point、同一center求值；單一值相等本身不是多項式相等的證明。",
        "en": "Recover every coefficient first to identify the whole polynomial,then evaluate at the same point and center.One coincident point value alone would not establish polynomial identity."
      },
      "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",
        "6",
        "8"
      ],
      [
        "2",
        "4",
        "12",
        "48"
      ],
      [
        "2",
        "4",
        "6",
        "8"
      ],
      "-12",
      "-4",
      "-4",
      true,
      "degree_at_most_n_polynomial_is_exactly_reconstructed"
    ]
  },
  "conclusion": {
    "canonical_value": "([2, 4, 6, 8], [2, 4, 12, 48], [2, 4, 6, 8], -12, -4, -4, 是, degree_at_most_n_polynomial_is_exactly_reconstructed)",
    "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": 52,
      "original_line_sha256": "e2655a6c7a8f8938fab381083567a32f1092d9ae7930f588effc909dac06dd08",
      "question_compact_sha256": "bb6f775406de6afccb42a723fef254a47bb54c53e631a481107cd5e3b6ce29bb",
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}