{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261015-advance-20261116-r03-taylor-remainder-theorem-0015",
  "pilot_number": "R0051-26465",
  "number": 26465,
  "family_id": "qf-mod47-taylor-qualification",
  "title": {
    "zh-Hant": "第 26465 題：Taylor餘項定理",
    "en": "Question 26465: An Exact Center Jet and Taylor Error Do Not Prove Interval-Wide Hypotheses"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261015-advance-20261116-r03-taylor-remainder-theorem-0015",
    "batch_id": "batch-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-taylor-remainder-theorem"
    ],
    "title_zh": "Taylor餘項定理",
    "stem_zh": "center、order、interval-wide n+1 differentiability與Taylor remainder責任。操作=taylor-qualification；資料={\"coefficients\":[16,14,12,10,8,6],\"center\":0,\"x\":[1,2],\"order\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "([16, 14, 12, 10, 8, 6], 0, 1/2, 5, [16, 14, 24, 60, 192, 720, 0], [16, 7, 3, 5/4, 1/2, 3/16], 447/16, 447/16, 0, interval_wide_n_plus_one_differentiability_is_separate_from_center_jet)",
      "explanation_zh": "Wave6第十層核對=([16, 14, 12, 10, 8, 6], 0, 1/2, 5, [16, 14, 24, 60, 192, 720, 0], [16, 7, 3, 5/4, 1/2, 3/16], 447/16, 447/16, 0, interval_wide_n_plus_one_differentiability_is_separate_from_center_jet)；依完整明示資料獨立重算；polynomial zero/exact Lagrange cases與sup-derivative bound分開；finite jet不證一般remainder、Taylor-series equality或analyticity。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 2,
      "skills": [
        "Taylor餘項定理"
      ]
    },
    "variant": {
      "family_id": "qf-mod47-taylor-qualification",
      "seed": 202611163015,
      "parameters": {
        "coefficients": [
          16,
          14,
          12,
          10,
          8,
          6
        ],
        "center": 0,
        "x": [
          1,
          2
        ],
        "order": 5,
        "domain": "Exact_polynomial_taylor_remainder",
        "operation": "taylor-qualification",
        "semantic_template": "qf-mod47-taylor-qualification/{\"center\":[0,1],\"coefficients\":[[16,1],[14,1],[12,1],[10,1],[8,1],[6,1]],\"order\":5,\"x\":[1,2]}"
      }
    },
    "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Banach／IVT／Taylor證明、MLE漸近性、factorization theorem、一般DAG iff或primitive-root主張、Lean或Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-15",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261015-advance-20261116-r03-taylor-remainder-theorem-0015",
    "batch_id": "batch-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "452d8b758b6b68b0a362c10fc2fc4bba96618290f9cb19df57f9581603e0b6b5",
      "family_id": "qf-mod47-taylor-qualification",
      "seed": 202611163015,
      "semantic_template": "qf-mod47-taylor-qualification/{\"center\":[0,1],\"coefficients\":[[16,1],[14,1],[12,1],[10,1],[8,1],[6,1]],\"order\":5,\"x\":[1,2]}"
    },
    "fields": {
      "title": "An Exact Center Jet and Taylor Error Do Not Prove Interval-Wide Hypotheses",
      "stem": "Coefficients=[16, 14, 12, 10, 8, 6] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=5; keep terms through that order, and define remainder=f(x)−Taylor_value. Give (coefficients, center, x, order, center_derivatives, Taylor_terms, Taylor_value, function_value, remainder, boundary).",
      "answer": {
        "kind": "expression",
        "value": "([16, 14, 12, 10, 8, 6], 0, 1/2, 5, [16, 14, 24, 60, 192, 720, 0], [16, 7, 3, 5/4, 1/2, 3/16], 447/16, 447/16, 0, interval_wide_n_plus_one_differentiability_is_separate_from_center_jet)",
        "explanation": "Retain center derivatives through order+1, every Taylor term, their sum, the exact function value and their difference. A finite jet at the center is not the general interval-wide differentiability or derivative-bound obligation. The exact result, retaining the requested order, is ([16, 14, 12, 10, 8, 6], 0, 1/2, 5, [16, 14, 24, 60, 192, 720, 0], [16, 7, 3, 5/4, 1/2, 3/16], 447/16, 447/16, 0, interval_wide_n_plus_one_differentiability_is_separate_from_center_jet)."
      },
      "skills": [
        "An Exact Center Jet and Taylor Error Do Not Prove Interval-Wide Hypotheses"
      ],
      "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": "center、order、interval-wide n+1 differentiability與Taylor remainder責任。操作=taylor-qualification；資料={\"coefficients\":[16,14,12,10,8,6],\"center\":0,\"x\":[1,2],\"order\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
        "en": "Coefficients=[16, 14, 12, 10, 8, 6] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=5; keep terms through that order, and define remainder=f(x)−Taylor_value. Give (coefficients, center, x, order, center_derivatives, Taylor_terms, Taylor_value, function_value, remainder, boundary)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source roles = [taylor-qualification]",
      "after": "derived = [[coefficients, center, x, order, center_derivatives, Taylor_terms, Taylor_value, function_value, remainder, boundary]]",
      "relation": "implies",
      "rule_id": "source_ordered_fields_without_JS",
      "conditions": [],
      "explanation": {
        "zh-Hant": "本頁直接列完整來源欄序，不需JavaScript；contract的6/4/6/6與IVT的8/7/8由原mode選，不補欄也不由答案形狀猜。",
        "en": "List the full source field order without JavaScript.Original modes select contract6/4/6/6 and IVT8/7/8 roles,without padding or answer-shape inference."
      },
      "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": "Taylor ORIGINAL x minus declared center := sub(1/2, 0)",
      "after": "Taylor ORIGINAL x minus declared center = 1/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor ORIGINAL x minus declared center，不以小數近似替代。",
          "en": "Recompute Taylor ORIGINAL x minus declared center with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "1/2",
          "0"
        ],
        "output": "1/2",
        "label": "Taylor ORIGINAL x minus declared center"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "(0)^0",
      "after": "Taylor ORIGINAL derivative0 center power0 term0 = 1",
      "relation": "implies",
      "rule_id": "integer_power",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以非負整數次有限乘積精確計算，零次為多項式中的乘法單位元。",
        "en": "Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative 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": {
        "operation": "integer_power",
        "inputs": [
          "0",
          "0"
        ],
        "output": "1",
        "label": "Taylor ORIGINAL derivative0 center power0 term0"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "Taylor ORIGINAL derivative0 falling factor times power term0 := mul(1, 1)",
      "after": "Taylor ORIGINAL derivative0 falling factor times power term0 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor ORIGINAL derivative0 falling factor times power term0，不以小數近似替代。",
          "en": "Recompute Taylor ORIGINAL derivative0 falling factor times power term0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "1"
        ],
        "output": "1",
        "label": "Taylor ORIGINAL derivative0 falling factor times power term0"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "Taylor ORIGINAL derivative0 ORIGINAL coefficient term0 := mul(16, 1)",
      "after": "Taylor ORIGINAL derivative0 ORIGINAL coefficient term0 = 16",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor ORIGINAL derivative0 ORIGINAL coefficient term0，不以小數近似替代。",
          "en": "Recompute Taylor ORIGINAL derivative0 ORIGINAL coefficient term0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "16",
          "1"
        ],
        "output": "16",
        "label": "Taylor ORIGINAL derivative0 ORIGINAL coefficient term0"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "Taylor ORIGINAL derivative0 ascending derivative sum term0 := sum(0, 16)",
      "after": "Taylor ORIGINAL derivative0 ascending derivative sum term0 = 16",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor ORIGINAL derivative0 ascending derivative sum term0，不以小數近似替代。",
          "en": "Recompute Taylor ORIGINAL derivative0 ascending derivative sum term0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "16"
        ],
        "output": "16",
        "label": "Taylor ORIGINAL derivative0 ascending derivative sum term0"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "(0)^1",
      "after": "Taylor ORIGINAL derivative0 center power1 term1 = 0",
      "relation": "implies",
      "rule_id": "integer_power",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以非負整數次有限乘積精確計算，零次為多項式中的乘法單位元。",
        "en": "Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative 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": {
        "operation": "integer_power",
        "inputs": [
          "0",
          "1"
        ],
        "output": "0",
        "label": "Taylor ORIGINAL derivative0 center power1 term1"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "Taylor ORIGINAL derivative0 falling factor times power term1 := mul(1, 0)",
      "after": "Taylor ORIGINAL derivative0 falling factor times power term1 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor ORIGINAL derivative0 falling factor times power term1，不以小數近似替代。",
          "en": "Recompute Taylor ORIGINAL derivative0 falling factor times power term1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "0"
        ],
        "output": "0",
        "label": "Taylor ORIGINAL derivative0 falling factor times power term1"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "Taylor ORIGINAL derivative0 ORIGINAL coefficient term1 := mul(14, 0)",
      "after": "Taylor ORIGINAL derivative0 ORIGINAL coefficient term1 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor ORIGINAL derivative0 ORIGINAL coefficient term1，不以小數近似替代。",
          "en": "Recompute Taylor ORIGINAL derivative0 ORIGINAL coefficient term1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "14",
          "0"
        ],
        "output": "0",
        "label": "Taylor ORIGINAL derivative0 ORIGINAL coefficient term1"
      }
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "Taylor ORIGINAL derivative0 ascending derivative sum term1 := sum(16, 0)",
      "after": "Taylor ORIGINAL derivative0 ascending derivative sum term1 = 16",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor ORIGINAL derivative0 ascending derivative sum term1，不以小數近似替代。",
          "en": "Recompute Taylor ORIGINAL derivative0 ascending derivative sum term1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "16",
          "0"
        ],
        "output": "16",
        "label": "Taylor ORIGINAL derivative0 ascending derivative sum term1"
      }
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "(0)^2",
      "after": "Taylor ORIGINAL derivative0 center power2 term2 = 0",
      "relation": "implies",
      "rule_id": "integer_power",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以非負整數次有限乘積精確計算，零次為多項式中的乘法單位元。",
        "en": "Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative 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": {
        "operation": "integer_power",
        "inputs": [
          "0",
          "2"
        ],
        "output": "0",
        "label": "Taylor ORIGINAL derivative0 center power2 term2"
      }
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "Taylor ORIGINAL derivative0 falling factor times power term2 := mul(1, 0)",
      "after": "Taylor ORIGINAL derivative0 falling factor times power term2 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor ORIGINAL derivative0 falling factor times power term2，不以小數近似替代。",
          "en": "Recompute Taylor ORIGINAL derivative0 falling factor times power term2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "0"
        ],
        "output": "0",
        "label": "Taylor ORIGINAL derivative0 falling factor times power term2"
      }
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "Taylor ORIGINAL derivative0 ORIGINAL coefficient term2 := mul(12, 0)",
      "after": "Taylor ORIGINAL derivative0 ORIGINAL coefficient term2 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor ORIGINAL derivative0 ORIGINAL coefficient term2，不以小數近似替代。",
          "en": "Recompute Taylor ORIGINAL derivative0 ORIGINAL coefficient term2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "12",
          "0"
        ],
        "output": "0",
        "label": "Taylor ORIGINAL derivative0 ORIGINAL coefficient term2"
      }
    },
    {
      "id": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "Taylor ORIGINAL derivative0 ascending derivative sum term2 := sum(16, 0)",
      "after": "Taylor ORIGINAL derivative0 ascending derivative sum term2 = 16",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor ORIGINAL derivative0 ascending derivative sum term2，不以小數近似替代。",
          "en": "Recompute Taylor ORIGINAL derivative0 ascending derivative sum term2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "16",
          "0"
        ],
        "output": "16",
        "label": "Taylor ORIGINAL derivative0 ascending derivative sum term2"
      }
    },
    {
      "id": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "(0)^3",
      "after": "Taylor ORIGINAL derivative0 center power3 term3 = 0",
      "relation": "implies",
      "rule_id": "integer_power",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以非負整數次有限乘積精確計算，零次為多項式中的乘法單位元。",
        "en": "Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative 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": {
        "operation": "integer_power",
        "inputs": [
          "0",
          "3"
        ],
        "output": "0",
        "label": "Taylor ORIGINAL derivative0 center power3 term3"
      }
    },
    {
      "id": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "Taylor ORIGINAL derivative0 falling factor times power term3 := mul(1, 0)",
      "after": "Taylor ORIGINAL derivative0 falling factor times power term3 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor ORIGINAL derivative0 falling factor times power term3，不以小數近似替代。",
          "en": "Recompute Taylor ORIGINAL derivative0 falling factor times power term3 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "0"
        ],
        "output": "0",
        "label": "Taylor ORIGINAL derivative0 falling factor times power term3"
      }
    },
    {
      "id": "S017",
      "uses": [
        "G001",
        "S016"
      ],
      "before": "Taylor ORIGINAL derivative0 ORIGINAL coefficient term3 := mul(10, 0)",
      "after": "Taylor ORIGINAL derivative0 ORIGINAL coefficient term3 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor ORIGINAL derivative0 ORIGINAL coefficient term3，不以小數近似替代。",
          "en": "Recompute Taylor ORIGINAL derivative0 ORIGINAL coefficient term3 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "10",
          "0"
        ],
        "output": "0",
        "label": "Taylor ORIGINAL derivative0 ORIGINAL coefficient term3"
      }
    },
    {
      "id": "S018",
      "uses": [
        "G001",
        "S017"
      ],
      "before": "Taylor ORIGINAL derivative0 ascending derivative sum term3 := sum(16, 0)",
      "after": "Taylor ORIGINAL derivative0 ascending derivative sum term3 = 16",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor ORIGINAL derivative0 ascending derivative sum term3，不以小數近似替代。",
          "en": "Recompute Taylor ORIGINAL derivative0 ascending derivative sum term3 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "16",
          "0"
        ],
        "output": "16",
        "label": "Taylor ORIGINAL derivative0 ascending derivative sum term3"
      }
    },
    {
      "id": "S019",
      "uses": [
        "G001",
        "S018"
      ],
      "before": "(0)^4",
      "after": "Taylor ORIGINAL derivative0 center power4 term4 = 0",
      "relation": "implies",
      "rule_id": "integer_power",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以非負整數次有限乘積精確計算，零次為多項式中的乘法單位元。",
        "en": "Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative 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": {
        "operation": "integer_power",
        "inputs": [
          "0",
          "4"
        ],
        "output": "0",
        "label": "Taylor ORIGINAL derivative0 center power4 term4"
      }
    },
    {
      "id": "S020",
      "uses": [
        "G001",
        "S019"
      ],
      "before": "Taylor ORIGINAL derivative0 falling factor times power term4 := mul(1, 0)",
      "after": "Taylor ORIGINAL derivative0 falling factor times power term4 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor ORIGINAL derivative0 falling factor times power term4，不以小數近似替代。",
          "en": "Recompute Taylor ORIGINAL derivative0 falling factor times power term4 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "0"
        ],
        "output": "0",
        "label": "Taylor ORIGINAL derivative0 falling factor times power term4"
      }
    },
    {
      "id": "S021",
      "uses": [
        "G001",
        "S020"
      ],
      "before": "Taylor ORIGINAL derivative0 ORIGINAL coefficient term4 := mul(8, 0)",
      "after": "Taylor ORIGINAL derivative0 ORIGINAL coefficient term4 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor ORIGINAL derivative0 ORIGINAL coefficient term4，不以小數近似替代。",
          "en": "Recompute Taylor ORIGINAL derivative0 ORIGINAL coefficient term4 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "8",
          "0"
        ],
        "output": "0",
        "label": "Taylor ORIGINAL derivative0 ORIGINAL coefficient term4"
      }
    },
    {
      "id": "S022",
      "uses": [
        "G001",
        "S021"
      ],
      "before": "Taylor ORIGINAL derivative0 ascending derivative sum term4 := sum(16, 0)",
      "after": "Taylor ORIGINAL derivative0 ascending derivative sum term4 = 16",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor ORIGINAL derivative0 ascending derivative sum term4，不以小數近似替代。",
          "en": "Recompute Taylor ORIGINAL derivative0 ascending derivative sum term4 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "16",
          "0"
        ],
        "output": "16",
        "label": "Taylor ORIGINAL derivative0 ascending derivative sum term4"
      }
    },
    {
      "id": "S023",
      "uses": [
        "G001",
        "S022"
      ],
      "before": "(0)^5",
      "after": "Taylor ORIGINAL derivative0 center power5 term5 = 0",
      "relation": "implies",
      "rule_id": "integer_power",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以非負整數次有限乘積精確計算，零次為多項式中的乘法單位元。",
        "en": "Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative 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": {
        "operation": "integer_power",
        "inputs": [
          "0",
          "5"
        ],
        "output": "0",
        "label": "Taylor ORIGINAL derivative0 center power5 term5"
      }
    },
    {
      "id": "S024",
      "uses": [
        "G001",
        "S023"
      ],
      "before": "Taylor ORIGINAL derivative0 falling factor times power term5 := mul(1, 0)",
      "after": "Taylor ORIGINAL derivative0 falling factor times power term5 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor ORIGINAL derivative0 falling factor times power term5，不以小數近似替代。",
          "en": "Recompute Taylor ORIGINAL derivative0 falling factor times power term5 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "0"
        ],
        "output": "0",
        "label": "Taylor ORIGINAL derivative0 falling factor times power term5"
      }
    },
    {
      "id": "S025",
      "uses": [
        "G001",
        "S024"
      ],
      "before": "Taylor ORIGINAL derivative0 ORIGINAL coefficient term5 := mul(6, 0)",
      "after": "Taylor ORIGINAL derivative0 ORIGINAL coefficient term5 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor ORIGINAL derivative0 ORIGINAL coefficient term5，不以小數近似替代。",
          "en": "Recompute Taylor ORIGINAL derivative0 ORIGINAL coefficient term5 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "6",
          "0"
        ],
        "output": "0",
        "label": "Taylor ORIGINAL derivative0 ORIGINAL coefficient term5"
      }
    },
    {
      "id": "S026",
      "uses": [
        "G001",
        "S025"
      ],
      "before": "Taylor ORIGINAL derivative0 ascending derivative sum term5 := sum(16, 0)",
      "after": "Taylor ORIGINAL derivative0 ascending derivative sum term5 = 16",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor ORIGINAL derivative0 ascending derivative sum term5，不以小數近似替代。",
          "en": "Recompute Taylor ORIGINAL derivative0 ascending derivative sum term5 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "16",
          "0"
        ],
        "output": "16",
        "label": "Taylor ORIGINAL derivative0 ascending derivative sum term5"
      }
    },
    {
      "id": "S027",
      "uses": [
        "G001",
        "S026"
      ],
      "before": "source roles = [0, 0, [16, 14, 12, 10, 8, 6]]",
      "after": "derived = [[[0, 16, 1, 1, 16], [1, 14, 1, 0, 0], [2, 12, 1, 0, 0], [3, 10, 1, 0, 0], [4, 8, 1, 0, 0], [5, 6, 1, 0, 0]], 16]",
      "relation": "implies",
      "rule_id": "Taylor_EACH_derivative_complete_falling_factorial_source_order",
      "conditions": [],
      "explanation": {
        "zh-Hant": "x^j的k階導數是j!/(j−k)!*x^(j−k)，j<k消失；逐原升冪次序加完，空和0。這裡整個polynomial已知，不只是中心jet。",
        "en": "The kth derivative of x^j is j!/(j−k)!*x^(j−k);terms j<k vanish.Sum the complete ascending-order terms,with empty sum0.The whole polynomial is known,not just a center jet."
      },
      "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": "S028",
      "uses": [
        "G001",
        "S027"
      ],
      "before": "(1/2)^0",
      "after": "Taylor ORIGINAL displacement power0 = 1",
      "relation": "implies",
      "rule_id": "integer_power",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以非負整數次有限乘積精確計算，零次為多項式中的乘法單位元。",
        "en": "Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative 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": {
        "operation": "integer_power",
        "inputs": [
          "1/2",
          "0"
        ],
        "output": "1",
        "label": "Taylor ORIGINAL displacement power0"
      }
    },
    {
      "id": "S029",
      "uses": [
        "G001",
        "S028"
      ],
      "before": "Taylor ORIGINAL derivative times displacement0 := mul(16, 1)",
      "after": "Taylor ORIGINAL derivative times displacement0 = 16",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor ORIGINAL derivative times displacement0，不以小數近似替代。",
          "en": "Recompute Taylor ORIGINAL derivative times displacement0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "16",
          "1"
        ],
        "output": "16",
        "label": "Taylor ORIGINAL derivative times displacement0"
      }
    },
    {
      "id": "S030",
      "uses": [
        "G001",
        "S029"
      ],
      "before": "Taylor ORIGINAL term0 divide factorial := div(16, 1)",
      "after": "Taylor ORIGINAL term0 divide factorial = 16",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor ORIGINAL term0 divide factorial，不以小數近似替代。",
          "en": "Recompute Taylor ORIGINAL term0 divide factorial with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "16",
          "1"
        ],
        "output": "16",
        "label": "Taylor ORIGINAL term0 divide factorial"
      }
    },
    {
      "id": "S031",
      "uses": [
        "G001",
        "S030"
      ],
      "before": "(0)^0",
      "after": "Taylor ORIGINAL derivative1 center power0 term1 = 1",
      "relation": "implies",
      "rule_id": "integer_power",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以非負整數次有限乘積精確計算，零次為多項式中的乘法單位元。",
        "en": "Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative 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": {
        "operation": "integer_power",
        "inputs": [
          "0",
          "0"
        ],
        "output": "1",
        "label": "Taylor ORIGINAL derivative1 center power0 term1"
      }
    },
    {
      "id": "S032",
      "uses": [
        "G001",
        "S031"
      ],
      "before": "Taylor ORIGINAL derivative1 falling factor times power term1 := mul(1, 1)",
      "after": "Taylor ORIGINAL derivative1 falling factor times power term1 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor ORIGINAL derivative1 falling factor times power term1，不以小數近似替代。",
          "en": "Recompute Taylor ORIGINAL derivative1 falling factor times power term1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "1"
        ],
        "output": "1",
        "label": "Taylor ORIGINAL derivative1 falling factor times power term1"
      }
    },
    {
      "id": "S033",
      "uses": [
        "G001",
        "S032"
      ],
      "before": "Taylor ORIGINAL derivative1 ORIGINAL coefficient term1 := mul(14, 1)",
      "after": "Taylor ORIGINAL derivative1 ORIGINAL coefficient term1 = 14",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor ORIGINAL derivative1 ORIGINAL coefficient term1，不以小數近似替代。",
          "en": "Recompute Taylor ORIGINAL derivative1 ORIGINAL coefficient term1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "14",
          "1"
        ],
        "output": "14",
        "label": "Taylor ORIGINAL derivative1 ORIGINAL coefficient term1"
      }
    },
    {
      "id": "S034",
      "uses": [
        "G001",
        "S033"
      ],
      "before": "Taylor ORIGINAL derivative1 ascending derivative sum term1 := sum(0, 14)",
      "after": "Taylor ORIGINAL derivative1 ascending derivative sum term1 = 14",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor ORIGINAL derivative1 ascending derivative sum term1，不以小數近似替代。",
          "en": "Recompute Taylor ORIGINAL derivative1 ascending derivative sum term1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "14"
        ],
        "output": "14",
        "label": "Taylor ORIGINAL derivative1 ascending derivative sum term1"
      }
    },
    {
      "id": "S035",
      "uses": [
        "G001",
        "S034"
      ],
      "before": "(0)^1",
      "after": "Taylor ORIGINAL derivative1 center power1 term2 = 0",
      "relation": "implies",
      "rule_id": "integer_power",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以非負整數次有限乘積精確計算，零次為多項式中的乘法單位元。",
        "en": "Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative 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": {
        "operation": "integer_power",
        "inputs": [
          "0",
          "1"
        ],
        "output": "0",
        "label": "Taylor ORIGINAL derivative1 center power1 term2"
      }
    },
    {
      "id": "S036",
      "uses": [
        "G001",
        "S035"
      ],
      "before": "Taylor ORIGINAL derivative1 falling factor times power term2 := mul(2, 0)",
      "after": "Taylor ORIGINAL derivative1 falling factor times power term2 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor ORIGINAL derivative1 falling factor times power term2，不以小數近似替代。",
          "en": "Recompute Taylor ORIGINAL derivative1 falling factor times power term2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "0"
        ],
        "output": "0",
        "label": "Taylor ORIGINAL derivative1 falling factor times power term2"
      }
    },
    {
      "id": "S037",
      "uses": [
        "G001",
        "S036"
      ],
      "before": "Taylor ORIGINAL derivative1 ORIGINAL coefficient term2 := mul(12, 0)",
      "after": "Taylor ORIGINAL derivative1 ORIGINAL coefficient term2 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor ORIGINAL derivative1 ORIGINAL coefficient term2，不以小數近似替代。",
          "en": "Recompute Taylor ORIGINAL derivative1 ORIGINAL coefficient term2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "12",
          "0"
        ],
        "output": "0",
        "label": "Taylor ORIGINAL derivative1 ORIGINAL coefficient term2"
      }
    },
    {
      "id": "S038",
      "uses": [
        "G001",
        "S037"
      ],
      "before": "Taylor ORIGINAL derivative1 ascending derivative sum term2 := sum(14, 0)",
      "after": "Taylor ORIGINAL derivative1 ascending derivative sum term2 = 14",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor ORIGINAL derivative1 ascending derivative sum term2，不以小數近似替代。",
          "en": "Recompute Taylor ORIGINAL derivative1 ascending derivative sum term2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "14",
          "0"
        ],
        "output": "14",
        "label": "Taylor ORIGINAL derivative1 ascending derivative sum term2"
      }
    },
    {
      "id": "S039",
      "uses": [
        "G001",
        "S038"
      ],
      "before": "(0)^2",
      "after": "Taylor ORIGINAL derivative1 center power2 term3 = 0",
      "relation": "implies",
      "rule_id": "integer_power",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以非負整數次有限乘積精確計算，零次為多項式中的乘法單位元。",
        "en": "Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative 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": {
        "operation": "integer_power",
        "inputs": [
          "0",
          "2"
        ],
        "output": "0",
        "label": "Taylor ORIGINAL derivative1 center power2 term3"
      }
    },
    {
      "id": "S040",
      "uses": [
        "G001",
        "S039"
      ],
      "before": "Taylor ORIGINAL derivative1 falling factor times power term3 := mul(3, 0)",
      "after": "Taylor ORIGINAL derivative1 falling factor times power term3 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor ORIGINAL derivative1 falling factor times power term3，不以小數近似替代。",
          "en": "Recompute Taylor ORIGINAL derivative1 falling factor times power term3 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "3",
          "0"
        ],
        "output": "0",
        "label": "Taylor ORIGINAL derivative1 falling factor times power term3"
      }
    },
    {
      "id": "S041",
      "uses": [
        "G001",
        "S040"
      ],
      "before": "Taylor ORIGINAL derivative1 ORIGINAL coefficient term3 := mul(10, 0)",
      "after": "Taylor ORIGINAL derivative1 ORIGINAL coefficient term3 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor ORIGINAL derivative1 ORIGINAL coefficient term3，不以小數近似替代。",
          "en": "Recompute Taylor ORIGINAL derivative1 ORIGINAL coefficient term3 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "10",
          "0"
        ],
        "output": "0",
        "label": "Taylor ORIGINAL derivative1 ORIGINAL coefficient term3"
      }
    },
    {
      "id": "S042",
      "uses": [
        "G001",
        "S041"
      ],
      "before": "Taylor ORIGINAL derivative1 ascending derivative sum term3 := sum(14, 0)",
      "after": "Taylor ORIGINAL derivative1 ascending derivative sum term3 = 14",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor ORIGINAL derivative1 ascending derivative sum term3，不以小數近似替代。",
          "en": "Recompute Taylor ORIGINAL derivative1 ascending derivative sum term3 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "14",
          "0"
        ],
        "output": "14",
        "label": "Taylor ORIGINAL derivative1 ascending derivative sum term3"
      }
    },
    {
      "id": "S043",
      "uses": [
        "G001",
        "S042"
      ],
      "before": "(0)^3",
      "after": "Taylor ORIGINAL derivative1 center power3 term4 = 0",
      "relation": "implies",
      "rule_id": "integer_power",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以非負整數次有限乘積精確計算，零次為多項式中的乘法單位元。",
        "en": "Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative 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": {
        "operation": "integer_power",
        "inputs": [
          "0",
          "3"
        ],
        "output": "0",
        "label": "Taylor ORIGINAL derivative1 center power3 term4"
      }
    },
    {
      "id": "S044",
      "uses": [
        "G001",
        "S043"
      ],
      "before": "Taylor ORIGINAL derivative1 falling factor times power term4 := mul(4, 0)",
      "after": "Taylor ORIGINAL derivative1 falling factor times power term4 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor ORIGINAL derivative1 falling factor times power term4，不以小數近似替代。",
          "en": "Recompute Taylor ORIGINAL derivative1 falling factor times power term4 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "4",
          "0"
        ],
        "output": "0",
        "label": "Taylor ORIGINAL derivative1 falling factor times power term4"
      }
    },
    {
      "id": "S045",
      "uses": [
        "G001",
        "S044"
      ],
      "before": "Taylor ORIGINAL derivative1 ORIGINAL coefficient term4 := mul(8, 0)",
      "after": "Taylor ORIGINAL derivative1 ORIGINAL coefficient term4 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor ORIGINAL derivative1 ORIGINAL coefficient term4，不以小數近似替代。",
          "en": "Recompute Taylor ORIGINAL derivative1 ORIGINAL coefficient term4 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "8",
          "0"
        ],
        "output": "0",
        "label": "Taylor ORIGINAL derivative1 ORIGINAL coefficient term4"
      }
    },
    {
      "id": "S046",
      "uses": [
        "G001",
        "S045"
      ],
      "before": "Taylor ORIGINAL derivative1 ascending derivative sum term4 := sum(14, 0)",
      "after": "Taylor ORIGINAL derivative1 ascending derivative sum term4 = 14",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor ORIGINAL derivative1 ascending derivative sum term4，不以小數近似替代。",
          "en": "Recompute Taylor ORIGINAL derivative1 ascending derivative sum term4 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "14",
          "0"
        ],
        "output": "14",
        "label": "Taylor ORIGINAL derivative1 ascending derivative sum term4"
      }
    },
    {
      "id": "S047",
      "uses": [
        "G001",
        "S046"
      ],
      "before": "(0)^4",
      "after": "Taylor ORIGINAL derivative1 center power4 term5 = 0",
      "relation": "implies",
      "rule_id": "integer_power",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以非負整數次有限乘積精確計算，零次為多項式中的乘法單位元。",
        "en": "Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative 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": {
        "operation": "integer_power",
        "inputs": [
          "0",
          "4"
        ],
        "output": "0",
        "label": "Taylor ORIGINAL derivative1 center power4 term5"
      }
    },
    {
      "id": "S048",
      "uses": [
        "G001",
        "S047"
      ],
      "before": "Taylor ORIGINAL derivative1 falling factor times power term5 := mul(5, 0)",
      "after": "Taylor ORIGINAL derivative1 falling factor times power term5 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor ORIGINAL derivative1 falling factor times power term5，不以小數近似替代。",
          "en": "Recompute Taylor ORIGINAL derivative1 falling factor times power term5 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "5",
          "0"
        ],
        "output": "0",
        "label": "Taylor ORIGINAL derivative1 falling factor times power term5"
      }
    },
    {
      "id": "S049",
      "uses": [
        "G001",
        "S048"
      ],
      "before": "Taylor ORIGINAL derivative1 ORIGINAL coefficient term5 := mul(6, 0)",
      "after": "Taylor ORIGINAL derivative1 ORIGINAL coefficient term5 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor ORIGINAL derivative1 ORIGINAL coefficient term5，不以小數近似替代。",
          "en": "Recompute Taylor ORIGINAL derivative1 ORIGINAL coefficient term5 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "6",
          "0"
        ],
        "output": "0",
        "label": "Taylor ORIGINAL derivative1 ORIGINAL coefficient term5"
      }
    },
    {
      "id": "S050",
      "uses": [
        "G001",
        "S049"
      ],
      "before": "Taylor ORIGINAL derivative1 ascending derivative sum term5 := sum(14, 0)",
      "after": "Taylor ORIGINAL derivative1 ascending derivative sum term5 = 14",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor ORIGINAL derivative1 ascending derivative sum term5，不以小數近似替代。",
          "en": "Recompute Taylor ORIGINAL derivative1 ascending derivative sum term5 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "14",
          "0"
        ],
        "output": "14",
        "label": "Taylor ORIGINAL derivative1 ascending derivative sum term5"
      }
    },
    {
      "id": "S051",
      "uses": [
        "G001",
        "S050"
      ],
      "before": "source roles = [1, 0, [16, 14, 12, 10, 8, 6]]",
      "after": "derived = [[[1, 14, 1, 1, 14], [2, 12, 2, 0, 0], [3, 10, 3, 0, 0], [4, 8, 4, 0, 0], [5, 6, 5, 0, 0]], 14]",
      "relation": "implies",
      "rule_id": "Taylor_EACH_derivative_complete_falling_factorial_source_order",
      "conditions": [],
      "explanation": {
        "zh-Hant": "x^j的k階導數是j!/(j−k)!*x^(j−k)，j<k消失；逐原升冪次序加完，空和0。這裡整個polynomial已知，不只是中心jet。",
        "en": "The kth derivative of x^j is j!/(j−k)!*x^(j−k);terms j<k vanish.Sum the complete ascending-order terms,with empty sum0.The whole polynomial is known,not just a center jet."
      },
      "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": "S052",
      "uses": [
        "G001",
        "S051"
      ],
      "before": "(1/2)^1",
      "after": "Taylor ORIGINAL displacement power1 = 1/2",
      "relation": "implies",
      "rule_id": "integer_power",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以非負整數次有限乘積精確計算，零次為多項式中的乘法單位元。",
        "en": "Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative 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": {
        "operation": "integer_power",
        "inputs": [
          "1/2",
          "1"
        ],
        "output": "1/2",
        "label": "Taylor ORIGINAL displacement power1"
      }
    },
    {
      "id": "S053",
      "uses": [
        "G001",
        "S052"
      ],
      "before": "Taylor ORIGINAL derivative times displacement1 := mul(14, 1/2)",
      "after": "Taylor ORIGINAL derivative times displacement1 = 7",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor ORIGINAL derivative times displacement1，不以小數近似替代。",
          "en": "Recompute Taylor ORIGINAL derivative times displacement1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "14",
          "1/2"
        ],
        "output": "7",
        "label": "Taylor ORIGINAL derivative times displacement1"
      }
    },
    {
      "id": "S054",
      "uses": [
        "G001",
        "S053"
      ],
      "before": "Taylor ORIGINAL term1 divide factorial := div(7, 1)",
      "after": "Taylor ORIGINAL term1 divide factorial = 7",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor ORIGINAL term1 divide factorial，不以小數近似替代。",
          "en": "Recompute Taylor ORIGINAL term1 divide factorial with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "7",
          "1"
        ],
        "output": "7",
        "label": "Taylor ORIGINAL term1 divide factorial"
      }
    },
    {
      "id": "S055",
      "uses": [
        "G001",
        "S054"
      ],
      "before": "(0)^0",
      "after": "Taylor ORIGINAL derivative2 center power0 term2 = 1",
      "relation": "implies",
      "rule_id": "integer_power",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以非負整數次有限乘積精確計算，零次為多項式中的乘法單位元。",
        "en": "Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative 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": {
        "operation": "integer_power",
        "inputs": [
          "0",
          "0"
        ],
        "output": "1",
        "label": "Taylor ORIGINAL derivative2 center power0 term2"
      }
    },
    {
      "id": "S056",
      "uses": [
        "G001",
        "S055"
      ],
      "before": "Taylor ORIGINAL derivative2 falling factor times power term2 := mul(2, 1)",
      "after": "Taylor ORIGINAL derivative2 falling factor times power term2 = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor ORIGINAL derivative2 falling factor times power term2，不以小數近似替代。",
          "en": "Recompute Taylor ORIGINAL derivative2 falling factor times power term2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "2",
          "1"
        ],
        "output": "2",
        "label": "Taylor ORIGINAL derivative2 falling factor times power term2"
      }
    },
    {
      "id": "S057",
      "uses": [
        "G001",
        "S056"
      ],
      "before": "Taylor ORIGINAL derivative2 ORIGINAL coefficient term2 := mul(12, 2)",
      "after": "Taylor ORIGINAL derivative2 ORIGINAL coefficient term2 = 24",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor ORIGINAL derivative2 ORIGINAL coefficient term2，不以小數近似替代。",
          "en": "Recompute Taylor ORIGINAL derivative2 ORIGINAL coefficient term2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "12",
          "2"
        ],
        "output": "24",
        "label": "Taylor ORIGINAL derivative2 ORIGINAL coefficient term2"
      }
    },
    {
      "id": "S058",
      "uses": [
        "G001",
        "S057"
      ],
      "before": "Taylor ORIGINAL derivative2 ascending derivative sum term2 := sum(0, 24)",
      "after": "Taylor ORIGINAL derivative2 ascending derivative sum term2 = 24",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor ORIGINAL derivative2 ascending derivative sum term2，不以小數近似替代。",
          "en": "Recompute Taylor ORIGINAL derivative2 ascending derivative sum term2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "24"
        ],
        "output": "24",
        "label": "Taylor ORIGINAL derivative2 ascending derivative sum term2"
      }
    },
    {
      "id": "S059",
      "uses": [
        "G001",
        "S058"
      ],
      "before": "(0)^1",
      "after": "Taylor ORIGINAL derivative2 center power1 term3 = 0",
      "relation": "implies",
      "rule_id": "integer_power",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以非負整數次有限乘積精確計算，零次為多項式中的乘法單位元。",
        "en": "Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative 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": {
        "operation": "integer_power",
        "inputs": [
          "0",
          "1"
        ],
        "output": "0",
        "label": "Taylor ORIGINAL derivative2 center power1 term3"
      }
    },
    {
      "id": "S060",
      "uses": [
        "G001",
        "S059"
      ],
      "before": "Taylor ORIGINAL derivative2 falling factor times power term3 := mul(6, 0)",
      "after": "Taylor ORIGINAL derivative2 falling factor times power term3 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor ORIGINAL derivative2 falling factor times power term3，不以小數近似替代。",
          "en": "Recompute Taylor ORIGINAL derivative2 falling factor times power term3 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "6",
          "0"
        ],
        "output": "0",
        "label": "Taylor ORIGINAL derivative2 falling factor times power term3"
      }
    },
    {
      "id": "S061",
      "uses": [
        "G001",
        "S060"
      ],
      "before": "Taylor ORIGINAL derivative2 ORIGINAL coefficient term3 := mul(10, 0)",
      "after": "Taylor ORIGINAL derivative2 ORIGINAL coefficient term3 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor ORIGINAL derivative2 ORIGINAL coefficient term3，不以小數近似替代。",
          "en": "Recompute Taylor ORIGINAL derivative2 ORIGINAL coefficient term3 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "10",
          "0"
        ],
        "output": "0",
        "label": "Taylor ORIGINAL derivative2 ORIGINAL coefficient term3"
      }
    },
    {
      "id": "S062",
      "uses": [
        "G001",
        "S061"
      ],
      "before": "Taylor ORIGINAL derivative2 ascending derivative sum term3 := sum(24, 0)",
      "after": "Taylor ORIGINAL derivative2 ascending derivative sum term3 = 24",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor ORIGINAL derivative2 ascending derivative sum term3，不以小數近似替代。",
          "en": "Recompute Taylor ORIGINAL derivative2 ascending derivative sum term3 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "24",
          "0"
        ],
        "output": "24",
        "label": "Taylor ORIGINAL derivative2 ascending derivative sum term3"
      }
    },
    {
      "id": "S063",
      "uses": [
        "G001",
        "S062"
      ],
      "before": "(0)^2",
      "after": "Taylor ORIGINAL derivative2 center power2 term4 = 0",
      "relation": "implies",
      "rule_id": "integer_power",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以非負整數次有限乘積精確計算，零次為多項式中的乘法單位元。",
        "en": "Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative 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": {
        "operation": "integer_power",
        "inputs": [
          "0",
          "2"
        ],
        "output": "0",
        "label": "Taylor ORIGINAL derivative2 center power2 term4"
      }
    },
    {
      "id": "S064",
      "uses": [
        "G001",
        "S063"
      ],
      "before": "Taylor ORIGINAL derivative2 falling factor times power term4 := mul(12, 0)",
      "after": "Taylor ORIGINAL derivative2 falling factor times power term4 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor ORIGINAL derivative2 falling factor times power term4，不以小數近似替代。",
          "en": "Recompute Taylor ORIGINAL derivative2 falling factor times power term4 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "12",
          "0"
        ],
        "output": "0",
        "label": "Taylor ORIGINAL derivative2 falling factor times power term4"
      }
    },
    {
      "id": "S065",
      "uses": [
        "G001",
        "S064"
      ],
      "before": "Taylor ORIGINAL derivative2 ORIGINAL coefficient term4 := mul(8, 0)",
      "after": "Taylor ORIGINAL derivative2 ORIGINAL coefficient term4 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor ORIGINAL derivative2 ORIGINAL coefficient term4，不以小數近似替代。",
          "en": "Recompute Taylor ORIGINAL derivative2 ORIGINAL coefficient term4 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "8",
          "0"
        ],
        "output": "0",
        "label": "Taylor ORIGINAL derivative2 ORIGINAL coefficient term4"
      }
    },
    {
      "id": "S066",
      "uses": [
        "G001",
        "S065"
      ],
      "before": "Taylor ORIGINAL derivative2 ascending derivative sum term4 := sum(24, 0)",
      "after": "Taylor ORIGINAL derivative2 ascending derivative sum term4 = 24",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor ORIGINAL derivative2 ascending derivative sum term4，不以小數近似替代。",
          "en": "Recompute Taylor ORIGINAL derivative2 ascending derivative sum term4 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "24",
          "0"
        ],
        "output": "24",
        "label": "Taylor ORIGINAL derivative2 ascending derivative sum term4"
      }
    },
    {
      "id": "S067",
      "uses": [
        "G001",
        "S066"
      ],
      "before": "(0)^3",
      "after": "Taylor ORIGINAL derivative2 center power3 term5 = 0",
      "relation": "implies",
      "rule_id": "integer_power",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以非負整數次有限乘積精確計算，零次為多項式中的乘法單位元。",
        "en": "Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative 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": {
        "operation": "integer_power",
        "inputs": [
          "0",
          "3"
        ],
        "output": "0",
        "label": "Taylor ORIGINAL derivative2 center power3 term5"
      }
    },
    {
      "id": "S068",
      "uses": [
        "G001",
        "S067"
      ],
      "before": "Taylor ORIGINAL derivative2 falling factor times power term5 := mul(20, 0)",
      "after": "Taylor ORIGINAL derivative2 falling factor times power term5 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor ORIGINAL derivative2 falling factor times power term5，不以小數近似替代。",
          "en": "Recompute Taylor ORIGINAL derivative2 falling factor times power term5 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "20",
          "0"
        ],
        "output": "0",
        "label": "Taylor ORIGINAL derivative2 falling factor times power term5"
      }
    },
    {
      "id": "S069",
      "uses": [
        "G001",
        "S068"
      ],
      "before": "Taylor ORIGINAL derivative2 ORIGINAL coefficient term5 := mul(6, 0)",
      "after": "Taylor ORIGINAL derivative2 ORIGINAL coefficient term5 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor ORIGINAL derivative2 ORIGINAL coefficient term5，不以小數近似替代。",
          "en": "Recompute Taylor ORIGINAL derivative2 ORIGINAL coefficient term5 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "6",
          "0"
        ],
        "output": "0",
        "label": "Taylor ORIGINAL derivative2 ORIGINAL coefficient term5"
      }
    },
    {
      "id": "S070",
      "uses": [
        "G001",
        "S069"
      ],
      "before": "Taylor ORIGINAL derivative2 ascending derivative sum term5 := sum(24, 0)",
      "after": "Taylor ORIGINAL derivative2 ascending derivative sum term5 = 24",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor ORIGINAL derivative2 ascending derivative sum term5，不以小數近似替代。",
          "en": "Recompute Taylor ORIGINAL derivative2 ascending derivative sum term5 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "24",
          "0"
        ],
        "output": "24",
        "label": "Taylor ORIGINAL derivative2 ascending derivative sum term5"
      }
    },
    {
      "id": "S071",
      "uses": [
        "G001",
        "S070"
      ],
      "before": "source roles = [2, 0, [16, 14, 12, 10, 8, 6]]",
      "after": "derived = [[[2, 12, 2, 1, 24], [3, 10, 6, 0, 0], [4, 8, 12, 0, 0], [5, 6, 20, 0, 0]], 24]",
      "relation": "implies",
      "rule_id": "Taylor_EACH_derivative_complete_falling_factorial_source_order",
      "conditions": [],
      "explanation": {
        "zh-Hant": "x^j的k階導數是j!/(j−k)!*x^(j−k)，j<k消失；逐原升冪次序加完，空和0。這裡整個polynomial已知，不只是中心jet。",
        "en": "The kth derivative of x^j is j!/(j−k)!*x^(j−k);terms j<k vanish.Sum the complete ascending-order terms,with empty sum0.The whole polynomial is known,not just a center jet."
      },
      "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": "S072",
      "uses": [
        "G001",
        "S071"
      ],
      "before": "(1/2)^2",
      "after": "Taylor ORIGINAL displacement power2 = 1/4",
      "relation": "implies",
      "rule_id": "integer_power",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以非負整數次有限乘積精確計算，零次為多項式中的乘法單位元。",
        "en": "Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative 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": {
        "operation": "integer_power",
        "inputs": [
          "1/2",
          "2"
        ],
        "output": "1/4",
        "label": "Taylor ORIGINAL displacement power2"
      }
    },
    {
      "id": "S073",
      "uses": [
        "G001",
        "S072"
      ],
      "before": "Taylor ORIGINAL derivative times displacement2 := mul(24, 1/4)",
      "after": "Taylor ORIGINAL derivative times displacement2 = 6",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor ORIGINAL derivative times displacement2，不以小數近似替代。",
          "en": "Recompute Taylor ORIGINAL derivative times displacement2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "24",
          "1/4"
        ],
        "output": "6",
        "label": "Taylor ORIGINAL derivative times displacement2"
      }
    },
    {
      "id": "S074",
      "uses": [
        "G001",
        "S073"
      ],
      "before": "Taylor ORIGINAL term2 divide factorial := div(6, 2)",
      "after": "Taylor ORIGINAL term2 divide factorial = 3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor ORIGINAL term2 divide factorial，不以小數近似替代。",
          "en": "Recompute Taylor ORIGINAL term2 divide factorial with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "6",
          "2"
        ],
        "output": "3",
        "label": "Taylor ORIGINAL term2 divide factorial"
      }
    },
    {
      "id": "S075",
      "uses": [
        "G001",
        "S074"
      ],
      "before": "(0)^0",
      "after": "Taylor ORIGINAL derivative3 center power0 term3 = 1",
      "relation": "implies",
      "rule_id": "integer_power",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以非負整數次有限乘積精確計算，零次為多項式中的乘法單位元。",
        "en": "Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative 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": {
        "operation": "integer_power",
        "inputs": [
          "0",
          "0"
        ],
        "output": "1",
        "label": "Taylor ORIGINAL derivative3 center power0 term3"
      }
    },
    {
      "id": "S076",
      "uses": [
        "G001",
        "S075"
      ],
      "before": "Taylor ORIGINAL derivative3 falling factor times power term3 := mul(6, 1)",
      "after": "Taylor ORIGINAL derivative3 falling factor times power term3 = 6",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor ORIGINAL derivative3 falling factor times power term3，不以小數近似替代。",
          "en": "Recompute Taylor ORIGINAL derivative3 falling factor times power term3 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "6",
          "1"
        ],
        "output": "6",
        "label": "Taylor ORIGINAL derivative3 falling factor times power term3"
      }
    },
    {
      "id": "S077",
      "uses": [
        "G001",
        "S076"
      ],
      "before": "Taylor ORIGINAL derivative3 ORIGINAL coefficient term3 := mul(10, 6)",
      "after": "Taylor ORIGINAL derivative3 ORIGINAL coefficient term3 = 60",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor ORIGINAL derivative3 ORIGINAL coefficient term3，不以小數近似替代。",
          "en": "Recompute Taylor ORIGINAL derivative3 ORIGINAL coefficient term3 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "10",
          "6"
        ],
        "output": "60",
        "label": "Taylor ORIGINAL derivative3 ORIGINAL coefficient term3"
      }
    },
    {
      "id": "S078",
      "uses": [
        "G001",
        "S077"
      ],
      "before": "Taylor ORIGINAL derivative3 ascending derivative sum term3 := sum(0, 60)",
      "after": "Taylor ORIGINAL derivative3 ascending derivative sum term3 = 60",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor ORIGINAL derivative3 ascending derivative sum term3，不以小數近似替代。",
          "en": "Recompute Taylor ORIGINAL derivative3 ascending derivative sum term3 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "60"
        ],
        "output": "60",
        "label": "Taylor ORIGINAL derivative3 ascending derivative sum term3"
      }
    },
    {
      "id": "S079",
      "uses": [
        "G001",
        "S078"
      ],
      "before": "(0)^1",
      "after": "Taylor ORIGINAL derivative3 center power1 term4 = 0",
      "relation": "implies",
      "rule_id": "integer_power",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以非負整數次有限乘積精確計算，零次為多項式中的乘法單位元。",
        "en": "Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative 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": {
        "operation": "integer_power",
        "inputs": [
          "0",
          "1"
        ],
        "output": "0",
        "label": "Taylor ORIGINAL derivative3 center power1 term4"
      }
    },
    {
      "id": "S080",
      "uses": [
        "G001",
        "S079"
      ],
      "before": "Taylor ORIGINAL derivative3 falling factor times power term4 := mul(24, 0)",
      "after": "Taylor ORIGINAL derivative3 falling factor times power term4 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor ORIGINAL derivative3 falling factor times power term4，不以小數近似替代。",
          "en": "Recompute Taylor ORIGINAL derivative3 falling factor times power term4 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "24",
          "0"
        ],
        "output": "0",
        "label": "Taylor ORIGINAL derivative3 falling factor times power term4"
      }
    },
    {
      "id": "S081",
      "uses": [
        "G001",
        "S080"
      ],
      "before": "Taylor ORIGINAL derivative3 ORIGINAL coefficient term4 := mul(8, 0)",
      "after": "Taylor ORIGINAL derivative3 ORIGINAL coefficient term4 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor ORIGINAL derivative3 ORIGINAL coefficient term4，不以小數近似替代。",
          "en": "Recompute Taylor ORIGINAL derivative3 ORIGINAL coefficient term4 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "8",
          "0"
        ],
        "output": "0",
        "label": "Taylor ORIGINAL derivative3 ORIGINAL coefficient term4"
      }
    },
    {
      "id": "S082",
      "uses": [
        "G001",
        "S081"
      ],
      "before": "Taylor ORIGINAL derivative3 ascending derivative sum term4 := sum(60, 0)",
      "after": "Taylor ORIGINAL derivative3 ascending derivative sum term4 = 60",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor ORIGINAL derivative3 ascending derivative sum term4，不以小數近似替代。",
          "en": "Recompute Taylor ORIGINAL derivative3 ascending derivative sum term4 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "60",
          "0"
        ],
        "output": "60",
        "label": "Taylor ORIGINAL derivative3 ascending derivative sum term4"
      }
    },
    {
      "id": "S083",
      "uses": [
        "G001",
        "S082"
      ],
      "before": "(0)^2",
      "after": "Taylor ORIGINAL derivative3 center power2 term5 = 0",
      "relation": "implies",
      "rule_id": "integer_power",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以非負整數次有限乘積精確計算，零次為多項式中的乘法單位元。",
        "en": "Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative 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": {
        "operation": "integer_power",
        "inputs": [
          "0",
          "2"
        ],
        "output": "0",
        "label": "Taylor ORIGINAL derivative3 center power2 term5"
      }
    },
    {
      "id": "S084",
      "uses": [
        "G001",
        "S083"
      ],
      "before": "Taylor ORIGINAL derivative3 falling factor times power term5 := mul(60, 0)",
      "after": "Taylor ORIGINAL derivative3 falling factor times power term5 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor ORIGINAL derivative3 falling factor times power term5，不以小數近似替代。",
          "en": "Recompute Taylor ORIGINAL derivative3 falling factor times power term5 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "60",
          "0"
        ],
        "output": "0",
        "label": "Taylor ORIGINAL derivative3 falling factor times power term5"
      }
    },
    {
      "id": "S085",
      "uses": [
        "G001",
        "S084"
      ],
      "before": "Taylor ORIGINAL derivative3 ORIGINAL coefficient term5 := mul(6, 0)",
      "after": "Taylor ORIGINAL derivative3 ORIGINAL coefficient term5 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor ORIGINAL derivative3 ORIGINAL coefficient term5，不以小數近似替代。",
          "en": "Recompute Taylor ORIGINAL derivative3 ORIGINAL coefficient term5 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "6",
          "0"
        ],
        "output": "0",
        "label": "Taylor ORIGINAL derivative3 ORIGINAL coefficient term5"
      }
    },
    {
      "id": "S086",
      "uses": [
        "G001",
        "S085"
      ],
      "before": "Taylor ORIGINAL derivative3 ascending derivative sum term5 := sum(60, 0)",
      "after": "Taylor ORIGINAL derivative3 ascending derivative sum term5 = 60",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor ORIGINAL derivative3 ascending derivative sum term5，不以小數近似替代。",
          "en": "Recompute Taylor ORIGINAL derivative3 ascending derivative sum term5 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "60",
          "0"
        ],
        "output": "60",
        "label": "Taylor ORIGINAL derivative3 ascending derivative sum term5"
      }
    },
    {
      "id": "S087",
      "uses": [
        "G001",
        "S086"
      ],
      "before": "source roles = [3, 0, [16, 14, 12, 10, 8, 6]]",
      "after": "derived = [[[3, 10, 6, 1, 60], [4, 8, 24, 0, 0], [5, 6, 60, 0, 0]], 60]",
      "relation": "implies",
      "rule_id": "Taylor_EACH_derivative_complete_falling_factorial_source_order",
      "conditions": [],
      "explanation": {
        "zh-Hant": "x^j的k階導數是j!/(j−k)!*x^(j−k)，j<k消失；逐原升冪次序加完，空和0。這裡整個polynomial已知，不只是中心jet。",
        "en": "The kth derivative of x^j is j!/(j−k)!*x^(j−k);terms j<k vanish.Sum the complete ascending-order terms,with empty sum0.The whole polynomial is known,not just a center jet."
      },
      "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": "S088",
      "uses": [
        "G001",
        "S087"
      ],
      "before": "(1/2)^3",
      "after": "Taylor ORIGINAL displacement power3 = 1/8",
      "relation": "implies",
      "rule_id": "integer_power",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以非負整數次有限乘積精確計算，零次為多項式中的乘法單位元。",
        "en": "Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative 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": {
        "operation": "integer_power",
        "inputs": [
          "1/2",
          "3"
        ],
        "output": "1/8",
        "label": "Taylor ORIGINAL displacement power3"
      }
    },
    {
      "id": "S089",
      "uses": [
        "G001",
        "S088"
      ],
      "before": "Taylor ORIGINAL derivative times displacement3 := mul(60, 1/8)",
      "after": "Taylor ORIGINAL derivative times displacement3 = 15/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor ORIGINAL derivative times displacement3，不以小數近似替代。",
          "en": "Recompute Taylor ORIGINAL derivative times displacement3 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "60",
          "1/8"
        ],
        "output": "15/2",
        "label": "Taylor ORIGINAL derivative times displacement3"
      }
    },
    {
      "id": "S090",
      "uses": [
        "G001",
        "S089"
      ],
      "before": "Taylor ORIGINAL term3 divide factorial := div(15/2, 6)",
      "after": "Taylor ORIGINAL term3 divide factorial = 5/4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor ORIGINAL term3 divide factorial，不以小數近似替代。",
          "en": "Recompute Taylor ORIGINAL term3 divide factorial with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "15/2",
          "6"
        ],
        "output": "5/4",
        "label": "Taylor ORIGINAL term3 divide factorial"
      }
    },
    {
      "id": "S091",
      "uses": [
        "G001",
        "S090"
      ],
      "before": "(0)^0",
      "after": "Taylor ORIGINAL derivative4 center power0 term4 = 1",
      "relation": "implies",
      "rule_id": "integer_power",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以非負整數次有限乘積精確計算，零次為多項式中的乘法單位元。",
        "en": "Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative 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": {
        "operation": "integer_power",
        "inputs": [
          "0",
          "0"
        ],
        "output": "1",
        "label": "Taylor ORIGINAL derivative4 center power0 term4"
      }
    },
    {
      "id": "S092",
      "uses": [
        "G001",
        "S091"
      ],
      "before": "Taylor ORIGINAL derivative4 falling factor times power term4 := mul(24, 1)",
      "after": "Taylor ORIGINAL derivative4 falling factor times power term4 = 24",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor ORIGINAL derivative4 falling factor times power term4，不以小數近似替代。",
          "en": "Recompute Taylor ORIGINAL derivative4 falling factor times power term4 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "24",
          "1"
        ],
        "output": "24",
        "label": "Taylor ORIGINAL derivative4 falling factor times power term4"
      }
    },
    {
      "id": "S093",
      "uses": [
        "G001",
        "S092"
      ],
      "before": "Taylor ORIGINAL derivative4 ORIGINAL coefficient term4 := mul(8, 24)",
      "after": "Taylor ORIGINAL derivative4 ORIGINAL coefficient term4 = 192",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor ORIGINAL derivative4 ORIGINAL coefficient term4，不以小數近似替代。",
          "en": "Recompute Taylor ORIGINAL derivative4 ORIGINAL coefficient term4 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "8",
          "24"
        ],
        "output": "192",
        "label": "Taylor ORIGINAL derivative4 ORIGINAL coefficient term4"
      }
    },
    {
      "id": "S094",
      "uses": [
        "G001",
        "S093"
      ],
      "before": "Taylor ORIGINAL derivative4 ascending derivative sum term4 := sum(0, 192)",
      "after": "Taylor ORIGINAL derivative4 ascending derivative sum term4 = 192",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor ORIGINAL derivative4 ascending derivative sum term4，不以小數近似替代。",
          "en": "Recompute Taylor ORIGINAL derivative4 ascending derivative sum term4 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "192"
        ],
        "output": "192",
        "label": "Taylor ORIGINAL derivative4 ascending derivative sum term4"
      }
    },
    {
      "id": "S095",
      "uses": [
        "G001",
        "S094"
      ],
      "before": "(0)^1",
      "after": "Taylor ORIGINAL derivative4 center power1 term5 = 0",
      "relation": "implies",
      "rule_id": "integer_power",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以非負整數次有限乘積精確計算，零次為多項式中的乘法單位元。",
        "en": "Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative 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": {
        "operation": "integer_power",
        "inputs": [
          "0",
          "1"
        ],
        "output": "0",
        "label": "Taylor ORIGINAL derivative4 center power1 term5"
      }
    },
    {
      "id": "S096",
      "uses": [
        "G001",
        "S095"
      ],
      "before": "Taylor ORIGINAL derivative4 falling factor times power term5 := mul(120, 0)",
      "after": "Taylor ORIGINAL derivative4 falling factor times power term5 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor ORIGINAL derivative4 falling factor times power term5，不以小數近似替代。",
          "en": "Recompute Taylor ORIGINAL derivative4 falling factor times power term5 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "120",
          "0"
        ],
        "output": "0",
        "label": "Taylor ORIGINAL derivative4 falling factor times power term5"
      }
    },
    {
      "id": "S097",
      "uses": [
        "G001",
        "S096"
      ],
      "before": "Taylor ORIGINAL derivative4 ORIGINAL coefficient term5 := mul(6, 0)",
      "after": "Taylor ORIGINAL derivative4 ORIGINAL coefficient term5 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor ORIGINAL derivative4 ORIGINAL coefficient term5，不以小數近似替代。",
          "en": "Recompute Taylor ORIGINAL derivative4 ORIGINAL coefficient term5 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "6",
          "0"
        ],
        "output": "0",
        "label": "Taylor ORIGINAL derivative4 ORIGINAL coefficient term5"
      }
    },
    {
      "id": "S098",
      "uses": [
        "G001",
        "S097"
      ],
      "before": "Taylor ORIGINAL derivative4 ascending derivative sum term5 := sum(192, 0)",
      "after": "Taylor ORIGINAL derivative4 ascending derivative sum term5 = 192",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor ORIGINAL derivative4 ascending derivative sum term5，不以小數近似替代。",
          "en": "Recompute Taylor ORIGINAL derivative4 ascending derivative sum term5 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "192",
          "0"
        ],
        "output": "192",
        "label": "Taylor ORIGINAL derivative4 ascending derivative sum term5"
      }
    },
    {
      "id": "S099",
      "uses": [
        "G001",
        "S098"
      ],
      "before": "source roles = [4, 0, [16, 14, 12, 10, 8, 6]]",
      "after": "derived = [[[4, 8, 24, 1, 192], [5, 6, 120, 0, 0]], 192]",
      "relation": "implies",
      "rule_id": "Taylor_EACH_derivative_complete_falling_factorial_source_order",
      "conditions": [],
      "explanation": {
        "zh-Hant": "x^j的k階導數是j!/(j−k)!*x^(j−k)，j<k消失；逐原升冪次序加完，空和0。這裡整個polynomial已知，不只是中心jet。",
        "en": "The kth derivative of x^j is j!/(j−k)!*x^(j−k);terms j<k vanish.Sum the complete ascending-order terms,with empty sum0.The whole polynomial is known,not just a center jet."
      },
      "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": "S100",
      "uses": [
        "G001",
        "S099"
      ],
      "before": "(1/2)^4",
      "after": "Taylor ORIGINAL displacement power4 = 1/16",
      "relation": "implies",
      "rule_id": "integer_power",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以非負整數次有限乘積精確計算，零次為多項式中的乘法單位元。",
        "en": "Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative 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": {
        "operation": "integer_power",
        "inputs": [
          "1/2",
          "4"
        ],
        "output": "1/16",
        "label": "Taylor ORIGINAL displacement power4"
      }
    },
    {
      "id": "S101",
      "uses": [
        "G001",
        "S100"
      ],
      "before": "Taylor ORIGINAL derivative times displacement4 := mul(192, 1/16)",
      "after": "Taylor ORIGINAL derivative times displacement4 = 12",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor ORIGINAL derivative times displacement4，不以小數近似替代。",
          "en": "Recompute Taylor ORIGINAL derivative times displacement4 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "192",
          "1/16"
        ],
        "output": "12",
        "label": "Taylor ORIGINAL derivative times displacement4"
      }
    },
    {
      "id": "S102",
      "uses": [
        "G001",
        "S101"
      ],
      "before": "Taylor ORIGINAL term4 divide factorial := div(12, 24)",
      "after": "Taylor ORIGINAL term4 divide factorial = 1/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor ORIGINAL term4 divide factorial，不以小數近似替代。",
          "en": "Recompute Taylor ORIGINAL term4 divide factorial with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "12",
          "24"
        ],
        "output": "1/2",
        "label": "Taylor ORIGINAL term4 divide factorial"
      }
    },
    {
      "id": "S103",
      "uses": [
        "G001",
        "S102"
      ],
      "before": "(0)^0",
      "after": "Taylor ORIGINAL derivative5 center power0 term5 = 1",
      "relation": "implies",
      "rule_id": "integer_power",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以非負整數次有限乘積精確計算，零次為多項式中的乘法單位元。",
        "en": "Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative 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": {
        "operation": "integer_power",
        "inputs": [
          "0",
          "0"
        ],
        "output": "1",
        "label": "Taylor ORIGINAL derivative5 center power0 term5"
      }
    },
    {
      "id": "S104",
      "uses": [
        "G001",
        "S103"
      ],
      "before": "Taylor ORIGINAL derivative5 falling factor times power term5 := mul(120, 1)",
      "after": "Taylor ORIGINAL derivative5 falling factor times power term5 = 120",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor ORIGINAL derivative5 falling factor times power term5，不以小數近似替代。",
          "en": "Recompute Taylor ORIGINAL derivative5 falling factor times power term5 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "120",
          "1"
        ],
        "output": "120",
        "label": "Taylor ORIGINAL derivative5 falling factor times power term5"
      }
    },
    {
      "id": "S105",
      "uses": [
        "G001",
        "S104"
      ],
      "before": "Taylor ORIGINAL derivative5 ORIGINAL coefficient term5 := mul(6, 120)",
      "after": "Taylor ORIGINAL derivative5 ORIGINAL coefficient term5 = 720",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor ORIGINAL derivative5 ORIGINAL coefficient term5，不以小數近似替代。",
          "en": "Recompute Taylor ORIGINAL derivative5 ORIGINAL coefficient term5 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "6",
          "120"
        ],
        "output": "720",
        "label": "Taylor ORIGINAL derivative5 ORIGINAL coefficient term5"
      }
    },
    {
      "id": "S106",
      "uses": [
        "G001",
        "S105"
      ],
      "before": "Taylor ORIGINAL derivative5 ascending derivative sum term5 := sum(0, 720)",
      "after": "Taylor ORIGINAL derivative5 ascending derivative sum term5 = 720",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor ORIGINAL derivative5 ascending derivative sum term5，不以小數近似替代。",
          "en": "Recompute Taylor ORIGINAL derivative5 ascending derivative sum term5 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "720"
        ],
        "output": "720",
        "label": "Taylor ORIGINAL derivative5 ascending derivative sum term5"
      }
    },
    {
      "id": "S107",
      "uses": [
        "G001",
        "S106"
      ],
      "before": "source roles = [5, 0, [16, 14, 12, 10, 8, 6]]",
      "after": "derived = [[[5, 6, 120, 1, 720]], 720]",
      "relation": "implies",
      "rule_id": "Taylor_EACH_derivative_complete_falling_factorial_source_order",
      "conditions": [],
      "explanation": {
        "zh-Hant": "x^j的k階導數是j!/(j−k)!*x^(j−k)，j<k消失；逐原升冪次序加完，空和0。這裡整個polynomial已知，不只是中心jet。",
        "en": "The kth derivative of x^j is j!/(j−k)!*x^(j−k);terms j<k vanish.Sum the complete ascending-order terms,with empty sum0.The whole polynomial is known,not just a center jet."
      },
      "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": "S108",
      "uses": [
        "G001",
        "S107"
      ],
      "before": "(1/2)^5",
      "after": "Taylor ORIGINAL displacement power5 = 1/32",
      "relation": "implies",
      "rule_id": "integer_power",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以非負整數次有限乘積精確計算，零次為多項式中的乘法單位元。",
        "en": "Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative 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": {
        "operation": "integer_power",
        "inputs": [
          "1/2",
          "5"
        ],
        "output": "1/32",
        "label": "Taylor ORIGINAL displacement power5"
      }
    },
    {
      "id": "S109",
      "uses": [
        "G001",
        "S108"
      ],
      "before": "Taylor ORIGINAL derivative times displacement5 := mul(720, 1/32)",
      "after": "Taylor ORIGINAL derivative times displacement5 = 45/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor ORIGINAL derivative times displacement5，不以小數近似替代。",
          "en": "Recompute Taylor ORIGINAL derivative times displacement5 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "720",
          "1/32"
        ],
        "output": "45/2",
        "label": "Taylor ORIGINAL derivative times displacement5"
      }
    },
    {
      "id": "S110",
      "uses": [
        "G001",
        "S109"
      ],
      "before": "Taylor ORIGINAL term5 divide factorial := div(45/2, 120)",
      "after": "Taylor ORIGINAL term5 divide factorial = 3/16",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor ORIGINAL term5 divide factorial，不以小數近似替代。",
          "en": "Recompute Taylor ORIGINAL term5 divide factorial with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "45/2",
          "120"
        ],
        "output": "3/16",
        "label": "Taylor ORIGINAL term5 divide factorial"
      }
    },
    {
      "id": "S111",
      "uses": [
        "G001",
        "S110"
      ],
      "before": "Taylor ORIGINAL partial sum0 := sum(0, 16)",
      "after": "Taylor ORIGINAL partial sum0 = 16",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor ORIGINAL partial sum0，不以小數近似替代。",
          "en": "Recompute Taylor ORIGINAL partial sum0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "16"
        ],
        "output": "16",
        "label": "Taylor ORIGINAL partial sum0"
      }
    },
    {
      "id": "S112",
      "uses": [
        "G001",
        "S111"
      ],
      "before": "Taylor ORIGINAL partial sum1 := sum(16, 7)",
      "after": "Taylor ORIGINAL partial sum1 = 23",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor ORIGINAL partial sum1，不以小數近似替代。",
          "en": "Recompute Taylor ORIGINAL partial sum1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "16",
          "7"
        ],
        "output": "23",
        "label": "Taylor ORIGINAL partial sum1"
      }
    },
    {
      "id": "S113",
      "uses": [
        "G001",
        "S112"
      ],
      "before": "Taylor ORIGINAL partial sum2 := sum(23, 3)",
      "after": "Taylor ORIGINAL partial sum2 = 26",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor ORIGINAL partial sum2，不以小數近似替代。",
          "en": "Recompute Taylor ORIGINAL partial sum2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "23",
          "3"
        ],
        "output": "26",
        "label": "Taylor ORIGINAL partial sum2"
      }
    },
    {
      "id": "S114",
      "uses": [
        "G001",
        "S113"
      ],
      "before": "Taylor ORIGINAL partial sum3 := sum(26, 5/4)",
      "after": "Taylor ORIGINAL partial sum3 = 109/4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor ORIGINAL partial sum3，不以小數近似替代。",
          "en": "Recompute Taylor ORIGINAL partial sum3 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "26",
          "5/4"
        ],
        "output": "109/4",
        "label": "Taylor ORIGINAL partial sum3"
      }
    },
    {
      "id": "S115",
      "uses": [
        "G001",
        "S114"
      ],
      "before": "Taylor ORIGINAL partial sum4 := sum(109/4, 1/2)",
      "after": "Taylor ORIGINAL partial sum4 = 111/4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor ORIGINAL partial sum4，不以小數近似替代。",
          "en": "Recompute Taylor ORIGINAL partial sum4 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "109/4",
          "1/2"
        ],
        "output": "111/4",
        "label": "Taylor ORIGINAL partial sum4"
      }
    },
    {
      "id": "S116",
      "uses": [
        "G001",
        "S115"
      ],
      "before": "Taylor ORIGINAL partial sum5 := sum(111/4, 3/16)",
      "after": "Taylor ORIGINAL partial sum5 = 447/16",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor ORIGINAL partial sum5，不以小數近似替代。",
          "en": "Recompute Taylor ORIGINAL partial sum5 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "111/4",
          "3/16"
        ],
        "output": "447/16",
        "label": "Taylor ORIGINAL partial sum5"
      }
    },
    {
      "id": "S117",
      "uses": [
        "G001",
        "S116"
      ],
      "before": "(1/2)^0",
      "after": "Taylor ORIGINAL full function at x ORIGINAL argument power0 = 1",
      "relation": "implies",
      "rule_id": "integer_power",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以非負整數次有限乘積精確計算，零次為多項式中的乘法單位元。",
        "en": "Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative 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": {
        "operation": "integer_power",
        "inputs": [
          "1/2",
          "0"
        ],
        "output": "1",
        "label": "Taylor ORIGINAL full function at x ORIGINAL argument power0"
      }
    },
    {
      "id": "S118",
      "uses": [
        "G001",
        "S117"
      ],
      "before": "Taylor ORIGINAL full function at x coefficient term0 := mul(16, 1)",
      "after": "Taylor ORIGINAL full function at x coefficient term0 = 16",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor ORIGINAL full function at x coefficient term0，不以小數近似替代。",
          "en": "Recompute Taylor ORIGINAL full function at x coefficient term0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "16",
          "1"
        ],
        "output": "16",
        "label": "Taylor ORIGINAL full function at x coefficient term0"
      }
    },
    {
      "id": "S119",
      "uses": [
        "G001",
        "S118"
      ],
      "before": "Taylor ORIGINAL full function at x ascending partial sum0 := sum(0, 16)",
      "after": "Taylor ORIGINAL full function at x ascending partial sum0 = 16",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor ORIGINAL full function at x ascending partial sum0，不以小數近似替代。",
          "en": "Recompute Taylor ORIGINAL full function at x ascending partial sum0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "16"
        ],
        "output": "16",
        "label": "Taylor ORIGINAL full function at x ascending partial sum0"
      }
    },
    {
      "id": "S120",
      "uses": [
        "G001",
        "S119"
      ],
      "before": "(1/2)^1",
      "after": "Taylor ORIGINAL full function at x ORIGINAL argument power1 = 1/2",
      "relation": "implies",
      "rule_id": "integer_power",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以非負整數次有限乘積精確計算，零次為多項式中的乘法單位元。",
        "en": "Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative 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": {
        "operation": "integer_power",
        "inputs": [
          "1/2",
          "1"
        ],
        "output": "1/2",
        "label": "Taylor ORIGINAL full function at x ORIGINAL argument power1"
      }
    },
    {
      "id": "S121",
      "uses": [
        "G001",
        "S120"
      ],
      "before": "Taylor ORIGINAL full function at x coefficient term1 := mul(14, 1/2)",
      "after": "Taylor ORIGINAL full function at x coefficient term1 = 7",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor ORIGINAL full function at x coefficient term1，不以小數近似替代。",
          "en": "Recompute Taylor ORIGINAL full function at x coefficient term1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "14",
          "1/2"
        ],
        "output": "7",
        "label": "Taylor ORIGINAL full function at x coefficient term1"
      }
    },
    {
      "id": "S122",
      "uses": [
        "G001",
        "S121"
      ],
      "before": "Taylor ORIGINAL full function at x ascending partial sum1 := sum(16, 7)",
      "after": "Taylor ORIGINAL full function at x ascending partial sum1 = 23",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor ORIGINAL full function at x ascending partial sum1，不以小數近似替代。",
          "en": "Recompute Taylor ORIGINAL full function at x ascending partial sum1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "16",
          "7"
        ],
        "output": "23",
        "label": "Taylor ORIGINAL full function at x ascending partial sum1"
      }
    },
    {
      "id": "S123",
      "uses": [
        "G001",
        "S122"
      ],
      "before": "(1/2)^2",
      "after": "Taylor ORIGINAL full function at x ORIGINAL argument power2 = 1/4",
      "relation": "implies",
      "rule_id": "integer_power",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以非負整數次有限乘積精確計算，零次為多項式中的乘法單位元。",
        "en": "Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative 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": {
        "operation": "integer_power",
        "inputs": [
          "1/2",
          "2"
        ],
        "output": "1/4",
        "label": "Taylor ORIGINAL full function at x ORIGINAL argument power2"
      }
    },
    {
      "id": "S124",
      "uses": [
        "G001",
        "S123"
      ],
      "before": "Taylor ORIGINAL full function at x coefficient term2 := mul(12, 1/4)",
      "after": "Taylor ORIGINAL full function at x coefficient term2 = 3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor ORIGINAL full function at x coefficient term2，不以小數近似替代。",
          "en": "Recompute Taylor ORIGINAL full function at x coefficient term2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "12",
          "1/4"
        ],
        "output": "3",
        "label": "Taylor ORIGINAL full function at x coefficient term2"
      }
    },
    {
      "id": "S125",
      "uses": [
        "G001",
        "S124"
      ],
      "before": "Taylor ORIGINAL full function at x ascending partial sum2 := sum(23, 3)",
      "after": "Taylor ORIGINAL full function at x ascending partial sum2 = 26",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor ORIGINAL full function at x ascending partial sum2，不以小數近似替代。",
          "en": "Recompute Taylor ORIGINAL full function at x ascending partial sum2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "23",
          "3"
        ],
        "output": "26",
        "label": "Taylor ORIGINAL full function at x ascending partial sum2"
      }
    },
    {
      "id": "S126",
      "uses": [
        "G001",
        "S125"
      ],
      "before": "(1/2)^3",
      "after": "Taylor ORIGINAL full function at x ORIGINAL argument power3 = 1/8",
      "relation": "implies",
      "rule_id": "integer_power",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以非負整數次有限乘積精確計算，零次為多項式中的乘法單位元。",
        "en": "Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative 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": {
        "operation": "integer_power",
        "inputs": [
          "1/2",
          "3"
        ],
        "output": "1/8",
        "label": "Taylor ORIGINAL full function at x ORIGINAL argument power3"
      }
    },
    {
      "id": "S127",
      "uses": [
        "G001",
        "S126"
      ],
      "before": "Taylor ORIGINAL full function at x coefficient term3 := mul(10, 1/8)",
      "after": "Taylor ORIGINAL full function at x coefficient term3 = 5/4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor ORIGINAL full function at x coefficient term3，不以小數近似替代。",
          "en": "Recompute Taylor ORIGINAL full function at x coefficient term3 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "10",
          "1/8"
        ],
        "output": "5/4",
        "label": "Taylor ORIGINAL full function at x coefficient term3"
      }
    },
    {
      "id": "S128",
      "uses": [
        "G001",
        "S127"
      ],
      "before": "Taylor ORIGINAL full function at x ascending partial sum3 := sum(26, 5/4)",
      "after": "Taylor ORIGINAL full function at x ascending partial sum3 = 109/4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor ORIGINAL full function at x ascending partial sum3，不以小數近似替代。",
          "en": "Recompute Taylor ORIGINAL full function at x ascending partial sum3 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "26",
          "5/4"
        ],
        "output": "109/4",
        "label": "Taylor ORIGINAL full function at x ascending partial sum3"
      }
    },
    {
      "id": "S129",
      "uses": [
        "G001",
        "S128"
      ],
      "before": "(1/2)^4",
      "after": "Taylor ORIGINAL full function at x ORIGINAL argument power4 = 1/16",
      "relation": "implies",
      "rule_id": "integer_power",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以非負整數次有限乘積精確計算，零次為多項式中的乘法單位元。",
        "en": "Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative 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": {
        "operation": "integer_power",
        "inputs": [
          "1/2",
          "4"
        ],
        "output": "1/16",
        "label": "Taylor ORIGINAL full function at x ORIGINAL argument power4"
      }
    },
    {
      "id": "S130",
      "uses": [
        "G001",
        "S129"
      ],
      "before": "Taylor ORIGINAL full function at x coefficient term4 := mul(8, 1/16)",
      "after": "Taylor ORIGINAL full function at x coefficient term4 = 1/2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor ORIGINAL full function at x coefficient term4，不以小數近似替代。",
          "en": "Recompute Taylor ORIGINAL full function at x coefficient term4 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "8",
          "1/16"
        ],
        "output": "1/2",
        "label": "Taylor ORIGINAL full function at x coefficient term4"
      }
    },
    {
      "id": "S131",
      "uses": [
        "G001",
        "S130"
      ],
      "before": "Taylor ORIGINAL full function at x ascending partial sum4 := sum(109/4, 1/2)",
      "after": "Taylor ORIGINAL full function at x ascending partial sum4 = 111/4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor ORIGINAL full function at x ascending partial sum4，不以小數近似替代。",
          "en": "Recompute Taylor ORIGINAL full function at x ascending partial sum4 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "109/4",
          "1/2"
        ],
        "output": "111/4",
        "label": "Taylor ORIGINAL full function at x ascending partial sum4"
      }
    },
    {
      "id": "S132",
      "uses": [
        "G001",
        "S131"
      ],
      "before": "(1/2)^5",
      "after": "Taylor ORIGINAL full function at x ORIGINAL argument power5 = 1/32",
      "relation": "implies",
      "rule_id": "integer_power",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以非負整數次有限乘積精確計算，零次為多項式中的乘法單位元。",
        "en": "Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative 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": {
        "operation": "integer_power",
        "inputs": [
          "1/2",
          "5"
        ],
        "output": "1/32",
        "label": "Taylor ORIGINAL full function at x ORIGINAL argument power5"
      }
    },
    {
      "id": "S133",
      "uses": [
        "G001",
        "S132"
      ],
      "before": "Taylor ORIGINAL full function at x coefficient term5 := mul(6, 1/32)",
      "after": "Taylor ORIGINAL full function at x coefficient term5 = 3/16",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor ORIGINAL full function at x coefficient term5，不以小數近似替代。",
          "en": "Recompute Taylor ORIGINAL full function at x coefficient term5 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "6",
          "1/32"
        ],
        "output": "3/16",
        "label": "Taylor ORIGINAL full function at x coefficient term5"
      }
    },
    {
      "id": "S134",
      "uses": [
        "G001",
        "S133"
      ],
      "before": "Taylor ORIGINAL full function at x ascending partial sum5 := sum(111/4, 3/16)",
      "after": "Taylor ORIGINAL full function at x ascending partial sum5 = 447/16",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor ORIGINAL full function at x ascending partial sum5，不以小數近似替代。",
          "en": "Recompute Taylor ORIGINAL full function at x ascending partial sum5 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "111/4",
          "3/16"
        ],
        "output": "447/16",
        "label": "Taylor ORIGINAL full function at x ascending partial sum5"
      }
    },
    {
      "id": "S135",
      "uses": [
        "G001",
        "S134"
      ],
      "before": "Taylor ORIGINAL f(x) minus Taylor value := sub(447/16, 447/16)",
      "after": "Taylor ORIGINAL f(x) minus Taylor value = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor ORIGINAL f(x) minus Taylor value，不以小數近似替代。",
          "en": "Recompute Taylor ORIGINAL f(x) minus Taylor value with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "447/16",
          "447/16"
        ],
        "output": "0",
        "label": "Taylor ORIGINAL f(x) minus Taylor value"
      }
    },
    {
      "id": "S136",
      "uses": [
        "G001",
        "S135"
      ],
      "before": "source roles = [[16, 14, 12, 10, 8, 6], 0, 1/2, 5, [16, 14, 24, 60, 192, 720], [16, 7, 3, 5/4, 1/2, 3/16]]",
      "after": "derived = [447/16, 447/16, 0, 5]",
      "relation": "implies",
      "rule_id": "Taylor_WHOLE_polynomial_shifted_basis_remainder_vs_interval_hypotheses",
      "conditions": [],
      "explanation": {
        "zh-Hant": "T_n=Σ f^(k)(a)(x−a)^k/k!，保含0階與所有requested zero terms；remainder=f(x)−T_n。polynomial可在整個interval微分，但一般函數不能由有限中心jet推出interval假設、series equality或analyticity。EMPTY係數degree−1不同於[0]degree0。",
        "en": "Use T_n=sum f^(k)(a)(x−a)^k/k!,including degree0 and all requested zero terms;remainder=f(x)−T_n.A polynomial differentiates on the whole interval,but a finite center jet of an arbitrary function does not supply interval hypotheses,series equality or analyticity.Empty coefficients have degree−1,distinct from[0]degree0."
      },
      "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": "S137",
      "uses": [
        "G001",
        "S136"
      ],
      "before": "Taylor ORIGINAL next derivative EMPTY derivative sum := sum()",
      "after": "Taylor ORIGINAL next derivative EMPTY derivative sum = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原始角色與已證中間量精確代值，不用近似；額外教學恆等式不另縮限原題域。",
        "en": "Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Taylor ORIGINAL next derivative EMPTY derivative sum，不以小數近似替代。",
          "en": "Recompute Taylor ORIGINAL next derivative EMPTY derivative sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [],
        "output": "0",
        "label": "Taylor ORIGINAL next derivative EMPTY derivative sum"
      }
    },
    {
      "id": "S138",
      "uses": [
        "G001",
        "S137"
      ],
      "before": "source roles = [6, 0, [16, 14, 12, 10, 8, 6]]",
      "after": "derived = [[], 0]",
      "relation": "implies",
      "rule_id": "Taylor_EACH_derivative_complete_falling_factorial_source_order",
      "conditions": [],
      "explanation": {
        "zh-Hant": "x^j的k階導數是j!/(j−k)!*x^(j−k)，j<k消失；逐原升冪次序加完，空和0。這裡整個polynomial已知，不只是中心jet。",
        "en": "The kth derivative of x^j is j!/(j−k)!*x^(j−k);terms j<k vanish.Sum the complete ascending-order terms,with empty sum0.The whole polynomial is known,not just a center jet."
      },
      "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": [
      [
        "16",
        "14",
        "12",
        "10",
        "8",
        "6"
      ],
      "0",
      "1/2",
      5,
      [
        "16",
        "14",
        "24",
        "60",
        "192",
        "720",
        "0"
      ],
      [
        "16",
        "7",
        "3",
        "5/4",
        "1/2",
        "3/16"
      ],
      "447/16",
      "447/16",
      "0",
      "interval_wide_n_plus_one_differentiability_is_separate_from_center_jet"
    ]
  },
  "conclusion": {
    "canonical_value": "([16, 14, 12, 10, 8, 6], 0, 1/2, 5, [16, 14, 24, 60, 192, 720, 0], [16, 7, 3, 5/4, 1/2, 3/16], 447/16, 447/16, 0, interval_wide_n_plus_one_differentiability_is_separate_from_center_jet)",
    "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-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations",
      "batch_path": "public/data/questions/batches/batch-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations/questions.jsonl",
      "batch_sha256": "141db31110896101a4d5602125b2fef56c1ba19eea0c90983156cb0f038b69d0",
      "original_line": 15,
      "original_line_sha256": "5a20fcf27df742cdf6f9faa9e4670535d5001a1b7fb9c40798b3835c88991b93",
      "question_compact_sha256": "452d8b758b6b68b0a362c10fc2fc4bba96618290f9cb19df57f9581603e0b6b5",
      "english_row_compact_sha256": "cd5971b391b3c85c2cd0dd564861527d4c47a28c35a4ac8e4ab5ce09bccb147d",
      "english_manifest_path": "public/data/i18n/editions/batch-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-taylor-remainder-theorem": {
        "sha256": "ee3585c22f12885c07e98aac923c9f9ee997c7ade4196f27b93655ce96498057",
        "english_sha256": "cee54894be892c0a0578aa77bb18bcc713fb0b5f6010d561fffe2552fa63f60f",
        "original_path": "public/data/mko/taylor-remainder-theorem.json",
        "english_path": "public/data/i18n/editions/mko-taylor-remainder-theorem.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "74f6f34e0a695e06f9d836b5f5c677f9f716d2291c75c8f2f5b809ff0935bcff",
    "original_candidate_sha256": "328fcb502a9a9af1b063131879971ba5870aafc76757d9b45b3a7117a95de143",
    "original_candidate_line_sha256": "328fcb502a9a9af1b063131879971ba5870aafc76757d9b45b3a7117a95de143"
  },
  "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-20261026-010-p52",
    "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": "90df1e889242e9afdf01e068788ee8500ef366ffcfc0496e30a6800de32631c3",
      "question_first_witness_sha256": "ae1e849608590ca856c9f3b6911ba58e3bf53c608eb7369606192a770f48bd5c"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 138,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "5e0f49362f9782295356fe6d9e31f19a9923bde0662e3612fdd46b5d78b4f589",
      "arithmetic_steps": 129
    },
    "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."
  }
}