{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261007-advance-20261008-r07-definite-integral-0063",
  "pilot_number": "R0064-32688",
  "number": 32688,
  "family_id": "qf-integral-polynomial",
  "title": {
    "zh-Hant": "第 32688 題：多項式的有向矩積分",
    "en": "Question 32688: Signed Moment Integral of a Polynomial"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261007-advance-20261008-r07-definite-integral-0063",
    "batch_id": "batch-20261007-advance-20261008-r07-definite-integral-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-definite-integral"
    ],
    "title_zh": "多項式的有向矩積分",
    "stem_zh": "全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[0, 13, -1, 1]。依明示多項式矩公式求從 a=-1 到 b=1 的有向積分，採最簡正分母分數。",
    "answer": {
      "kind": "expression",
      "value": "-2/3",
      "explanation_zh": "有向矩積分=-2/3。只在多項式矩切片內以 canonical primitive 端點差獨立計算；反向端點改號、相同端點為零，不宣稱一般 FTC 或一般面積理論已完成。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 2,
      "skills": [
        "多項式的有向矩積分"
      ]
    },
    "variant": {
      "family_id": "qf-integral-polynomial",
      "seed": 202610087063,
      "parameters": {
        "f": [
          0,
          13,
          -1,
          1
        ],
        "a": -1,
        "b": 1,
        "operation": "polynomial_integral",
        "domain": "reals",
        "representation": "ascending_powers",
        "semantic_template": "qf-integral-polynomial/{\"inputs\":[[\"0\",\"13\",\"-1\",\"1\"],\"-1\",\"1\"],\"operation\":\"polynomial_integral\"}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_integer_rational_arithmetic",
        "producer_finite_enumeration",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-07",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261007-advance-20261008-r07-definite-integral-0063",
    "batch_id": "batch-20261007-advance-20261008-r07-definite-integral-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "6cf64cadfd90d7591e82ea125c9cd7c00bc72358b8d0616419963a74c417f1bf",
      "family_id": "qf-integral-polynomial",
      "seed": 202610087063,
      "semantic_template": "qf-integral-polynomial/{\"inputs\":[[\"0\",\"13\",\"-1\",\"1\"],\"-1\",\"1\"],\"operation\":\"polynomial_integral\"}"
    },
    "fields": {
      "title": "Signed Moment Integral of a Polynomial",
      "stem": "On all R, f has ascending coefficients [0, 13, -1, 1], starting with its constant term. Use coefficientwise exact signed moments on oriented endpoints a=-1, b=1. Give a reduced fraction with positive denominator, retaining /1 for integers.",
      "answer": {
        "kind": "expression",
        "value": "-2/3",
        "explanation": "Sum a_k*(b^(k+1)−a^(k+1))/(k+1) exactly to obtain -2/3. Keep orientation and sign; restricted polynomial moment checks do not prove a general FTC bridge."
      },
      "skills": [
        "Signed Moment Integral of a Polynomial"
      ],
      "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": "全實數論域，以下函數／點均以明示實數模型解讀。升冪係數 f=[0, 13, -1, 1]。依明示多項式矩公式求從 a=-1 到 b=1 的有向積分，採最簡正分母分數。",
        "en": "On all R, f has ascending coefficients [0, 13, -1, 1], starting with its constant term. Use coefficientwise exact signed moments on oriented endpoints a=-1, b=1. Give a reduced fraction with positive denominator, retaining /1 for integers."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source roles = [qf-integral-polynomial, polynomial_integral, reals, []]",
      "after": "derived = [[signed_moment_fraction_RETAIN1]]",
      "relation": "implies",
      "rule_id": "R64_SOURCE_exact_legacy_interface_BASE_metadata_AND_output_roles",
      "conditions": [],
      "explanation": {
        "zh-Hant": "legacy核來源精確欄位及operation/domain；BASE整數保metadata但不當oracle。先分NUMBER、字串、fractionpair或vector等式，不強套同一格式；原中英／AST／status不變。",
        "en": "Legacy banks enforce their exact source keys/operation/domain.BASE integers retain metadata without treating it as an oracle.Distinguish NUMBER,string,fraction pair and vector equality formats;retain original bilingual data/AST/status."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。",
          "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "source roles = [[0, 13, -1, 1]]",
      "after": "derived = [[0, 13, -1, 1]]",
      "relation": "implies",
      "rule_id": "Moment_RAW_ascending_1to4_coefficients_BEFORE_trim_ALL_validated",
      "conditions": [],
      "explanation": {
        "zh-Hant": "raw係數1..4升冪，全部SAFE先驗再trim尾0、零多項式留[0]；a=b或全0也不略過raw欄位。",
        "en": "RAW ascending coefficient arity1..4 and ALL SAFE values are checked BEFORE trimming trailing zeros,retaining[0].Equal endpoints/zero polynomials do not waive RAW validation."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。",
          "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "Polynomial ORIGINAL coefficient0 lower endpoint ORIGINAL power multiplication1 := mul(1, -1)",
      "after": "Polynomial ORIGINAL coefficient0 lower endpoint ORIGINAL power multiplication1 = -1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Polynomial ORIGINAL coefficient0 lower endpoint ORIGINAL power multiplication1，不以小數近似替代。",
          "en": "Recompute Polynomial ORIGINAL coefficient0 lower endpoint ORIGINAL power multiplication1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "-1"
        ],
        "output": "-1",
        "label": "Polynomial ORIGINAL coefficient0 lower endpoint ORIGINAL power multiplication1"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "source roles = [Polynomial ORIGINAL coefficient0 lower endpoint, -1, 1]",
      "after": "derived = [-1]",
      "relation": "implies",
      "rule_id": "Moment_INTEGER_power_complete_factors_NO_raw_recap",
      "conditions": [],
      "explanation": {
        "zh-Hant": "明列整數冪的全部乘法，保負底數的奇偶符號；衍生值不重新當raw SAFE，也不用浮點pow。",
        "en": "Show EVERY integer-power multiplication,retaining parity signs for negative bases.Derived values are not RAW-SAFE recapped or computed by floating pow."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。",
          "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "Polynomial ORIGINAL coefficient0 upper endpoint ORIGINAL power multiplication1 := mul(1, 1)",
      "after": "Polynomial ORIGINAL coefficient0 upper endpoint ORIGINAL power multiplication1 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Polynomial ORIGINAL coefficient0 upper endpoint ORIGINAL power multiplication1，不以小數近似替代。",
          "en": "Recompute Polynomial ORIGINAL coefficient0 upper endpoint ORIGINAL power multiplication1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "1"
        ],
        "output": "1",
        "label": "Polynomial ORIGINAL coefficient0 upper endpoint ORIGINAL power multiplication1"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "source roles = [Polynomial ORIGINAL coefficient0 upper endpoint, 1, 1]",
      "after": "derived = [1]",
      "relation": "implies",
      "rule_id": "Moment_INTEGER_power_complete_factors_NO_raw_recap",
      "conditions": [],
      "explanation": {
        "zh-Hant": "明列整數冪的全部乘法，保負底數的奇偶符號；衍生值不重新當raw SAFE，也不用浮點pow。",
        "en": "Show EVERY integer-power multiplication,retaining parity signs for negative bases.Derived values are not RAW-SAFE recapped or computed by floating pow."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。",
          "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "Polynomial ORIGINAL coefficient0 ORIGINAL upper power minus lower power := sub(1, -1)",
      "after": "Polynomial ORIGINAL coefficient0 ORIGINAL upper power minus lower power = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Polynomial ORIGINAL coefficient0 ORIGINAL upper power minus lower power，不以小數近似替代。",
          "en": "Recompute Polynomial ORIGINAL coefficient0 ORIGINAL upper power minus lower power with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "1",
          "-1"
        ],
        "output": "2",
        "label": "Polynomial ORIGINAL coefficient0 ORIGINAL upper power minus lower power"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "Polynomial ORIGINAL coefficient0 ORIGINAL coefficient times power difference := mul(0, 2)",
      "after": "Polynomial ORIGINAL coefficient0 ORIGINAL coefficient times power difference = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Polynomial ORIGINAL coefficient0 ORIGINAL coefficient times power difference，不以小數近似替代。",
          "en": "Recompute Polynomial ORIGINAL coefficient0 ORIGINAL coefficient times power difference with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "2"
        ],
        "output": "0",
        "label": "Polynomial ORIGINAL coefficient0 ORIGINAL coefficient times power difference"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "Polynomial ORIGINAL coefficient0 ORIGINAL signed moment over positive exponent := div(0, 1)",
      "after": "Polynomial ORIGINAL coefficient0 ORIGINAL signed moment over positive exponent = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Polynomial ORIGINAL coefficient0 ORIGINAL signed moment over positive exponent，不以小數近似替代。",
          "en": "Recompute Polynomial ORIGINAL coefficient0 ORIGINAL signed moment over positive exponent with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "0",
          "1"
        ],
        "output": "0",
        "label": "Polynomial ORIGINAL coefficient0 ORIGINAL signed moment over positive exponent"
      }
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "source roles = [Polynomial ORIGINAL coefficient0, 0, 0, -1, 1]",
      "after": "derived = [-1, 1, 2, 0/1]",
      "relation": "implies",
      "rule_id": "Moment_GIVEN_signed_polynomial_coefficient_moment_orientation",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依明示有號多項式矩c_k(b^(k+1)−a^(k+1))/(k+1)精確算。a>b合法，符號不取絕對值；這是受限矩模型，不把算術核對當一般FTC／可積性證書。",
        "en": "Use the DECLARED signed polynomial moment c_k(b^(k+1)−a^(k+1))/(k+1)exactly.Reversed endpoints are legal;do not take absolute values.This restricted moment model is not general FTC/integrability Evidence."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。",
          "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "Polynomial moment ORIGINAL sum through coefficient0 := sum(0, 0)",
      "after": "Polynomial moment ORIGINAL sum through coefficient0 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Polynomial moment ORIGINAL sum through coefficient0，不以小數近似替代。",
          "en": "Recompute Polynomial moment ORIGINAL sum through coefficient0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "0"
        ],
        "output": "0",
        "label": "Polynomial moment ORIGINAL sum through coefficient0"
      }
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "Polynomial ORIGINAL coefficient1 lower endpoint ORIGINAL power multiplication1 := mul(1, -1)",
      "after": "Polynomial ORIGINAL coefficient1 lower endpoint ORIGINAL power multiplication1 = -1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Polynomial ORIGINAL coefficient1 lower endpoint ORIGINAL power multiplication1，不以小數近似替代。",
          "en": "Recompute Polynomial ORIGINAL coefficient1 lower endpoint ORIGINAL power multiplication1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "-1"
        ],
        "output": "-1",
        "label": "Polynomial ORIGINAL coefficient1 lower endpoint ORIGINAL power multiplication1"
      }
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "Polynomial ORIGINAL coefficient1 lower endpoint ORIGINAL power multiplication2 := mul(-1, -1)",
      "after": "Polynomial ORIGINAL coefficient1 lower endpoint ORIGINAL power multiplication2 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Polynomial ORIGINAL coefficient1 lower endpoint ORIGINAL power multiplication2，不以小數近似替代。",
          "en": "Recompute Polynomial ORIGINAL coefficient1 lower endpoint ORIGINAL power multiplication2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-1",
          "-1"
        ],
        "output": "1",
        "label": "Polynomial ORIGINAL coefficient1 lower endpoint ORIGINAL power multiplication2"
      }
    },
    {
      "id": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "source roles = [Polynomial ORIGINAL coefficient1 lower endpoint, -1, 2]",
      "after": "derived = [1]",
      "relation": "implies",
      "rule_id": "Moment_INTEGER_power_complete_factors_NO_raw_recap",
      "conditions": [],
      "explanation": {
        "zh-Hant": "明列整數冪的全部乘法，保負底數的奇偶符號；衍生值不重新當raw SAFE，也不用浮點pow。",
        "en": "Show EVERY integer-power multiplication,retaining parity signs for negative bases.Derived values are not RAW-SAFE recapped or computed by floating pow."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。",
          "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "Polynomial ORIGINAL coefficient1 upper endpoint ORIGINAL power multiplication1 := mul(1, 1)",
      "after": "Polynomial ORIGINAL coefficient1 upper endpoint ORIGINAL power multiplication1 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Polynomial ORIGINAL coefficient1 upper endpoint ORIGINAL power multiplication1，不以小數近似替代。",
          "en": "Recompute Polynomial ORIGINAL coefficient1 upper endpoint ORIGINAL power multiplication1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "1"
        ],
        "output": "1",
        "label": "Polynomial ORIGINAL coefficient1 upper endpoint ORIGINAL power multiplication1"
      }
    },
    {
      "id": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "Polynomial ORIGINAL coefficient1 upper endpoint ORIGINAL power multiplication2 := mul(1, 1)",
      "after": "Polynomial ORIGINAL coefficient1 upper endpoint ORIGINAL power multiplication2 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Polynomial ORIGINAL coefficient1 upper endpoint ORIGINAL power multiplication2，不以小數近似替代。",
          "en": "Recompute Polynomial ORIGINAL coefficient1 upper endpoint ORIGINAL power multiplication2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "1"
        ],
        "output": "1",
        "label": "Polynomial ORIGINAL coefficient1 upper endpoint ORIGINAL power multiplication2"
      }
    },
    {
      "id": "S017",
      "uses": [
        "G001",
        "S016"
      ],
      "before": "source roles = [Polynomial ORIGINAL coefficient1 upper endpoint, 1, 2]",
      "after": "derived = [1]",
      "relation": "implies",
      "rule_id": "Moment_INTEGER_power_complete_factors_NO_raw_recap",
      "conditions": [],
      "explanation": {
        "zh-Hant": "明列整數冪的全部乘法，保負底數的奇偶符號；衍生值不重新當raw SAFE，也不用浮點pow。",
        "en": "Show EVERY integer-power multiplication,retaining parity signs for negative bases.Derived values are not RAW-SAFE recapped or computed by floating pow."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。",
          "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S018",
      "uses": [
        "G001",
        "S017"
      ],
      "before": "Polynomial ORIGINAL coefficient1 ORIGINAL upper power minus lower power := sub(1, 1)",
      "after": "Polynomial ORIGINAL coefficient1 ORIGINAL upper power minus lower power = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Polynomial ORIGINAL coefficient1 ORIGINAL upper power minus lower power，不以小數近似替代。",
          "en": "Recompute Polynomial ORIGINAL coefficient1 ORIGINAL upper power minus lower power with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "1",
          "1"
        ],
        "output": "0",
        "label": "Polynomial ORIGINAL coefficient1 ORIGINAL upper power minus lower power"
      }
    },
    {
      "id": "S019",
      "uses": [
        "G001",
        "S018"
      ],
      "before": "Polynomial ORIGINAL coefficient1 ORIGINAL coefficient times power difference := mul(13, 0)",
      "after": "Polynomial ORIGINAL coefficient1 ORIGINAL coefficient times power difference = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Polynomial ORIGINAL coefficient1 ORIGINAL coefficient times power difference，不以小數近似替代。",
          "en": "Recompute Polynomial ORIGINAL coefficient1 ORIGINAL coefficient times power difference with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "13",
          "0"
        ],
        "output": "0",
        "label": "Polynomial ORIGINAL coefficient1 ORIGINAL coefficient times power difference"
      }
    },
    {
      "id": "S020",
      "uses": [
        "G001",
        "S019"
      ],
      "before": "Polynomial ORIGINAL coefficient1 ORIGINAL signed moment over positive exponent := div(0, 2)",
      "after": "Polynomial ORIGINAL coefficient1 ORIGINAL signed moment over positive exponent = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Polynomial ORIGINAL coefficient1 ORIGINAL signed moment over positive exponent，不以小數近似替代。",
          "en": "Recompute Polynomial ORIGINAL coefficient1 ORIGINAL signed moment over positive exponent with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "0",
          "2"
        ],
        "output": "0",
        "label": "Polynomial ORIGINAL coefficient1 ORIGINAL signed moment over positive exponent"
      }
    },
    {
      "id": "S021",
      "uses": [
        "G001",
        "S020"
      ],
      "before": "source roles = [Polynomial ORIGINAL coefficient1, 13, 1, -1, 1]",
      "after": "derived = [1, 1, 0, 0/1]",
      "relation": "implies",
      "rule_id": "Moment_GIVEN_signed_polynomial_coefficient_moment_orientation",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依明示有號多項式矩c_k(b^(k+1)−a^(k+1))/(k+1)精確算。a>b合法，符號不取絕對值；這是受限矩模型，不把算術核對當一般FTC／可積性證書。",
        "en": "Use the DECLARED signed polynomial moment c_k(b^(k+1)−a^(k+1))/(k+1)exactly.Reversed endpoints are legal;do not take absolute values.This restricted moment model is not general FTC/integrability Evidence."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。",
          "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S022",
      "uses": [
        "G001",
        "S021"
      ],
      "before": "Polynomial moment ORIGINAL sum through coefficient1 := sum(0, 0)",
      "after": "Polynomial moment ORIGINAL sum through coefficient1 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Polynomial moment ORIGINAL sum through coefficient1，不以小數近似替代。",
          "en": "Recompute Polynomial moment ORIGINAL sum through coefficient1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "0"
        ],
        "output": "0",
        "label": "Polynomial moment ORIGINAL sum through coefficient1"
      }
    },
    {
      "id": "S023",
      "uses": [
        "G001",
        "S022"
      ],
      "before": "Polynomial ORIGINAL coefficient2 lower endpoint ORIGINAL power multiplication1 := mul(1, -1)",
      "after": "Polynomial ORIGINAL coefficient2 lower endpoint ORIGINAL power multiplication1 = -1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Polynomial ORIGINAL coefficient2 lower endpoint ORIGINAL power multiplication1，不以小數近似替代。",
          "en": "Recompute Polynomial ORIGINAL coefficient2 lower endpoint ORIGINAL power multiplication1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "-1"
        ],
        "output": "-1",
        "label": "Polynomial ORIGINAL coefficient2 lower endpoint ORIGINAL power multiplication1"
      }
    },
    {
      "id": "S024",
      "uses": [
        "G001",
        "S023"
      ],
      "before": "Polynomial ORIGINAL coefficient2 lower endpoint ORIGINAL power multiplication2 := mul(-1, -1)",
      "after": "Polynomial ORIGINAL coefficient2 lower endpoint ORIGINAL power multiplication2 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Polynomial ORIGINAL coefficient2 lower endpoint ORIGINAL power multiplication2，不以小數近似替代。",
          "en": "Recompute Polynomial ORIGINAL coefficient2 lower endpoint ORIGINAL power multiplication2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-1",
          "-1"
        ],
        "output": "1",
        "label": "Polynomial ORIGINAL coefficient2 lower endpoint ORIGINAL power multiplication2"
      }
    },
    {
      "id": "S025",
      "uses": [
        "G001",
        "S024"
      ],
      "before": "Polynomial ORIGINAL coefficient2 lower endpoint ORIGINAL power multiplication3 := mul(1, -1)",
      "after": "Polynomial ORIGINAL coefficient2 lower endpoint ORIGINAL power multiplication3 = -1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Polynomial ORIGINAL coefficient2 lower endpoint ORIGINAL power multiplication3，不以小數近似替代。",
          "en": "Recompute Polynomial ORIGINAL coefficient2 lower endpoint ORIGINAL power multiplication3 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "-1"
        ],
        "output": "-1",
        "label": "Polynomial ORIGINAL coefficient2 lower endpoint ORIGINAL power multiplication3"
      }
    },
    {
      "id": "S026",
      "uses": [
        "G001",
        "S025"
      ],
      "before": "source roles = [Polynomial ORIGINAL coefficient2 lower endpoint, -1, 3]",
      "after": "derived = [-1]",
      "relation": "implies",
      "rule_id": "Moment_INTEGER_power_complete_factors_NO_raw_recap",
      "conditions": [],
      "explanation": {
        "zh-Hant": "明列整數冪的全部乘法，保負底數的奇偶符號；衍生值不重新當raw SAFE，也不用浮點pow。",
        "en": "Show EVERY integer-power multiplication,retaining parity signs for negative bases.Derived values are not RAW-SAFE recapped or computed by floating pow."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。",
          "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S027",
      "uses": [
        "G001",
        "S026"
      ],
      "before": "Polynomial ORIGINAL coefficient2 upper endpoint ORIGINAL power multiplication1 := mul(1, 1)",
      "after": "Polynomial ORIGINAL coefficient2 upper endpoint ORIGINAL power multiplication1 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Polynomial ORIGINAL coefficient2 upper endpoint ORIGINAL power multiplication1，不以小數近似替代。",
          "en": "Recompute Polynomial ORIGINAL coefficient2 upper endpoint ORIGINAL power multiplication1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "1"
        ],
        "output": "1",
        "label": "Polynomial ORIGINAL coefficient2 upper endpoint ORIGINAL power multiplication1"
      }
    },
    {
      "id": "S028",
      "uses": [
        "G001",
        "S027"
      ],
      "before": "Polynomial ORIGINAL coefficient2 upper endpoint ORIGINAL power multiplication2 := mul(1, 1)",
      "after": "Polynomial ORIGINAL coefficient2 upper endpoint ORIGINAL power multiplication2 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Polynomial ORIGINAL coefficient2 upper endpoint ORIGINAL power multiplication2，不以小數近似替代。",
          "en": "Recompute Polynomial ORIGINAL coefficient2 upper endpoint ORIGINAL power multiplication2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "1"
        ],
        "output": "1",
        "label": "Polynomial ORIGINAL coefficient2 upper endpoint ORIGINAL power multiplication2"
      }
    },
    {
      "id": "S029",
      "uses": [
        "G001",
        "S028"
      ],
      "before": "Polynomial ORIGINAL coefficient2 upper endpoint ORIGINAL power multiplication3 := mul(1, 1)",
      "after": "Polynomial ORIGINAL coefficient2 upper endpoint ORIGINAL power multiplication3 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Polynomial ORIGINAL coefficient2 upper endpoint ORIGINAL power multiplication3，不以小數近似替代。",
          "en": "Recompute Polynomial ORIGINAL coefficient2 upper endpoint ORIGINAL power multiplication3 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "1"
        ],
        "output": "1",
        "label": "Polynomial ORIGINAL coefficient2 upper endpoint ORIGINAL power multiplication3"
      }
    },
    {
      "id": "S030",
      "uses": [
        "G001",
        "S029"
      ],
      "before": "source roles = [Polynomial ORIGINAL coefficient2 upper endpoint, 1, 3]",
      "after": "derived = [1]",
      "relation": "implies",
      "rule_id": "Moment_INTEGER_power_complete_factors_NO_raw_recap",
      "conditions": [],
      "explanation": {
        "zh-Hant": "明列整數冪的全部乘法，保負底數的奇偶符號；衍生值不重新當raw SAFE，也不用浮點pow。",
        "en": "Show EVERY integer-power multiplication,retaining parity signs for negative bases.Derived values are not RAW-SAFE recapped or computed by floating pow."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。",
          "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S031",
      "uses": [
        "G001",
        "S030"
      ],
      "before": "Polynomial ORIGINAL coefficient2 ORIGINAL upper power minus lower power := sub(1, -1)",
      "after": "Polynomial ORIGINAL coefficient2 ORIGINAL upper power minus lower power = 2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Polynomial ORIGINAL coefficient2 ORIGINAL upper power minus lower power，不以小數近似替代。",
          "en": "Recompute Polynomial ORIGINAL coefficient2 ORIGINAL upper power minus lower power with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "1",
          "-1"
        ],
        "output": "2",
        "label": "Polynomial ORIGINAL coefficient2 ORIGINAL upper power minus lower power"
      }
    },
    {
      "id": "S032",
      "uses": [
        "G001",
        "S031"
      ],
      "before": "Polynomial ORIGINAL coefficient2 ORIGINAL coefficient times power difference := mul(-1, 2)",
      "after": "Polynomial ORIGINAL coefficient2 ORIGINAL coefficient times power difference = -2",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Polynomial ORIGINAL coefficient2 ORIGINAL coefficient times power difference，不以小數近似替代。",
          "en": "Recompute Polynomial ORIGINAL coefficient2 ORIGINAL coefficient times power difference with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-1",
          "2"
        ],
        "output": "-2",
        "label": "Polynomial ORIGINAL coefficient2 ORIGINAL coefficient times power difference"
      }
    },
    {
      "id": "S033",
      "uses": [
        "G001",
        "S032"
      ],
      "before": "Polynomial ORIGINAL coefficient2 ORIGINAL signed moment over positive exponent := div(-2, 3)",
      "after": "Polynomial ORIGINAL coefficient2 ORIGINAL signed moment over positive exponent = -2/3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Polynomial ORIGINAL coefficient2 ORIGINAL signed moment over positive exponent，不以小數近似替代。",
          "en": "Recompute Polynomial ORIGINAL coefficient2 ORIGINAL signed moment over positive exponent with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "-2",
          "3"
        ],
        "output": "-2/3",
        "label": "Polynomial ORIGINAL coefficient2 ORIGINAL signed moment over positive exponent"
      }
    },
    {
      "id": "S034",
      "uses": [
        "G001",
        "S033"
      ],
      "before": "source roles = [Polynomial ORIGINAL coefficient2, -1, 2, -1, 1]",
      "after": "derived = [-1, 1, 2, -2/3]",
      "relation": "implies",
      "rule_id": "Moment_GIVEN_signed_polynomial_coefficient_moment_orientation",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依明示有號多項式矩c_k(b^(k+1)−a^(k+1))/(k+1)精確算。a>b合法，符號不取絕對值；這是受限矩模型，不把算術核對當一般FTC／可積性證書。",
        "en": "Use the DECLARED signed polynomial moment c_k(b^(k+1)−a^(k+1))/(k+1)exactly.Reversed endpoints are legal;do not take absolute values.This restricted moment model is not general FTC/integrability Evidence."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。",
          "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S035",
      "uses": [
        "G001",
        "S034"
      ],
      "before": "Polynomial moment ORIGINAL sum through coefficient2 := sum(0, -2/3)",
      "after": "Polynomial moment ORIGINAL sum through coefficient2 = -2/3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Polynomial moment ORIGINAL sum through coefficient2，不以小數近似替代。",
          "en": "Recompute Polynomial moment ORIGINAL sum through coefficient2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "-2/3"
        ],
        "output": "-2/3",
        "label": "Polynomial moment ORIGINAL sum through coefficient2"
      }
    },
    {
      "id": "S036",
      "uses": [
        "G001",
        "S035"
      ],
      "before": "Polynomial ORIGINAL coefficient3 lower endpoint ORIGINAL power multiplication1 := mul(1, -1)",
      "after": "Polynomial ORIGINAL coefficient3 lower endpoint ORIGINAL power multiplication1 = -1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Polynomial ORIGINAL coefficient3 lower endpoint ORIGINAL power multiplication1，不以小數近似替代。",
          "en": "Recompute Polynomial ORIGINAL coefficient3 lower endpoint ORIGINAL power multiplication1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "-1"
        ],
        "output": "-1",
        "label": "Polynomial ORIGINAL coefficient3 lower endpoint ORIGINAL power multiplication1"
      }
    },
    {
      "id": "S037",
      "uses": [
        "G001",
        "S036"
      ],
      "before": "Polynomial ORIGINAL coefficient3 lower endpoint ORIGINAL power multiplication2 := mul(-1, -1)",
      "after": "Polynomial ORIGINAL coefficient3 lower endpoint ORIGINAL power multiplication2 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Polynomial ORIGINAL coefficient3 lower endpoint ORIGINAL power multiplication2，不以小數近似替代。",
          "en": "Recompute Polynomial ORIGINAL coefficient3 lower endpoint ORIGINAL power multiplication2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-1",
          "-1"
        ],
        "output": "1",
        "label": "Polynomial ORIGINAL coefficient3 lower endpoint ORIGINAL power multiplication2"
      }
    },
    {
      "id": "S038",
      "uses": [
        "G001",
        "S037"
      ],
      "before": "Polynomial ORIGINAL coefficient3 lower endpoint ORIGINAL power multiplication3 := mul(1, -1)",
      "after": "Polynomial ORIGINAL coefficient3 lower endpoint ORIGINAL power multiplication3 = -1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Polynomial ORIGINAL coefficient3 lower endpoint ORIGINAL power multiplication3，不以小數近似替代。",
          "en": "Recompute Polynomial ORIGINAL coefficient3 lower endpoint ORIGINAL power multiplication3 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "-1"
        ],
        "output": "-1",
        "label": "Polynomial ORIGINAL coefficient3 lower endpoint ORIGINAL power multiplication3"
      }
    },
    {
      "id": "S039",
      "uses": [
        "G001",
        "S038"
      ],
      "before": "Polynomial ORIGINAL coefficient3 lower endpoint ORIGINAL power multiplication4 := mul(-1, -1)",
      "after": "Polynomial ORIGINAL coefficient3 lower endpoint ORIGINAL power multiplication4 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Polynomial ORIGINAL coefficient3 lower endpoint ORIGINAL power multiplication4，不以小數近似替代。",
          "en": "Recompute Polynomial ORIGINAL coefficient3 lower endpoint ORIGINAL power multiplication4 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-1",
          "-1"
        ],
        "output": "1",
        "label": "Polynomial ORIGINAL coefficient3 lower endpoint ORIGINAL power multiplication4"
      }
    },
    {
      "id": "S040",
      "uses": [
        "G001",
        "S039"
      ],
      "before": "source roles = [Polynomial ORIGINAL coefficient3 lower endpoint, -1, 4]",
      "after": "derived = [1]",
      "relation": "implies",
      "rule_id": "Moment_INTEGER_power_complete_factors_NO_raw_recap",
      "conditions": [],
      "explanation": {
        "zh-Hant": "明列整數冪的全部乘法，保負底數的奇偶符號；衍生值不重新當raw SAFE，也不用浮點pow。",
        "en": "Show EVERY integer-power multiplication,retaining parity signs for negative bases.Derived values are not RAW-SAFE recapped or computed by floating pow."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。",
          "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S041",
      "uses": [
        "G001",
        "S040"
      ],
      "before": "Polynomial ORIGINAL coefficient3 upper endpoint ORIGINAL power multiplication1 := mul(1, 1)",
      "after": "Polynomial ORIGINAL coefficient3 upper endpoint ORIGINAL power multiplication1 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Polynomial ORIGINAL coefficient3 upper endpoint ORIGINAL power multiplication1，不以小數近似替代。",
          "en": "Recompute Polynomial ORIGINAL coefficient3 upper endpoint ORIGINAL power multiplication1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "1"
        ],
        "output": "1",
        "label": "Polynomial ORIGINAL coefficient3 upper endpoint ORIGINAL power multiplication1"
      }
    },
    {
      "id": "S042",
      "uses": [
        "G001",
        "S041"
      ],
      "before": "Polynomial ORIGINAL coefficient3 upper endpoint ORIGINAL power multiplication2 := mul(1, 1)",
      "after": "Polynomial ORIGINAL coefficient3 upper endpoint ORIGINAL power multiplication2 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Polynomial ORIGINAL coefficient3 upper endpoint ORIGINAL power multiplication2，不以小數近似替代。",
          "en": "Recompute Polynomial ORIGINAL coefficient3 upper endpoint ORIGINAL power multiplication2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "1"
        ],
        "output": "1",
        "label": "Polynomial ORIGINAL coefficient3 upper endpoint ORIGINAL power multiplication2"
      }
    },
    {
      "id": "S043",
      "uses": [
        "G001",
        "S042"
      ],
      "before": "Polynomial ORIGINAL coefficient3 upper endpoint ORIGINAL power multiplication3 := mul(1, 1)",
      "after": "Polynomial ORIGINAL coefficient3 upper endpoint ORIGINAL power multiplication3 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Polynomial ORIGINAL coefficient3 upper endpoint ORIGINAL power multiplication3，不以小數近似替代。",
          "en": "Recompute Polynomial ORIGINAL coefficient3 upper endpoint ORIGINAL power multiplication3 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "1"
        ],
        "output": "1",
        "label": "Polynomial ORIGINAL coefficient3 upper endpoint ORIGINAL power multiplication3"
      }
    },
    {
      "id": "S044",
      "uses": [
        "G001",
        "S043"
      ],
      "before": "Polynomial ORIGINAL coefficient3 upper endpoint ORIGINAL power multiplication4 := mul(1, 1)",
      "after": "Polynomial ORIGINAL coefficient3 upper endpoint ORIGINAL power multiplication4 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Polynomial ORIGINAL coefficient3 upper endpoint ORIGINAL power multiplication4，不以小數近似替代。",
          "en": "Recompute Polynomial ORIGINAL coefficient3 upper endpoint ORIGINAL power multiplication4 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "1"
        ],
        "output": "1",
        "label": "Polynomial ORIGINAL coefficient3 upper endpoint ORIGINAL power multiplication4"
      }
    },
    {
      "id": "S045",
      "uses": [
        "G001",
        "S044"
      ],
      "before": "source roles = [Polynomial ORIGINAL coefficient3 upper endpoint, 1, 4]",
      "after": "derived = [1]",
      "relation": "implies",
      "rule_id": "Moment_INTEGER_power_complete_factors_NO_raw_recap",
      "conditions": [],
      "explanation": {
        "zh-Hant": "明列整數冪的全部乘法，保負底數的奇偶符號；衍生值不重新當raw SAFE，也不用浮點pow。",
        "en": "Show EVERY integer-power multiplication,retaining parity signs for negative bases.Derived values are not RAW-SAFE recapped or computed by floating pow."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。",
          "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S046",
      "uses": [
        "G001",
        "S045"
      ],
      "before": "Polynomial ORIGINAL coefficient3 ORIGINAL upper power minus lower power := sub(1, 1)",
      "after": "Polynomial ORIGINAL coefficient3 ORIGINAL upper power minus lower power = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Polynomial ORIGINAL coefficient3 ORIGINAL upper power minus lower power，不以小數近似替代。",
          "en": "Recompute Polynomial ORIGINAL coefficient3 ORIGINAL upper power minus lower power with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "1",
          "1"
        ],
        "output": "0",
        "label": "Polynomial ORIGINAL coefficient3 ORIGINAL upper power minus lower power"
      }
    },
    {
      "id": "S047",
      "uses": [
        "G001",
        "S046"
      ],
      "before": "Polynomial ORIGINAL coefficient3 ORIGINAL coefficient times power difference := mul(1, 0)",
      "after": "Polynomial ORIGINAL coefficient3 ORIGINAL coefficient times power difference = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Polynomial ORIGINAL coefficient3 ORIGINAL coefficient times power difference，不以小數近似替代。",
          "en": "Recompute Polynomial ORIGINAL coefficient3 ORIGINAL coefficient times power difference with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "0"
        ],
        "output": "0",
        "label": "Polynomial ORIGINAL coefficient3 ORIGINAL coefficient times power difference"
      }
    },
    {
      "id": "S048",
      "uses": [
        "G001",
        "S047"
      ],
      "before": "Polynomial ORIGINAL coefficient3 ORIGINAL signed moment over positive exponent := div(0, 4)",
      "after": "Polynomial ORIGINAL coefficient3 ORIGINAL signed moment over positive exponent = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Polynomial ORIGINAL coefficient3 ORIGINAL signed moment over positive exponent，不以小數近似替代。",
          "en": "Recompute Polynomial ORIGINAL coefficient3 ORIGINAL signed moment over positive exponent with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "0",
          "4"
        ],
        "output": "0",
        "label": "Polynomial ORIGINAL coefficient3 ORIGINAL signed moment over positive exponent"
      }
    },
    {
      "id": "S049",
      "uses": [
        "G001",
        "S048"
      ],
      "before": "source roles = [Polynomial ORIGINAL coefficient3, 1, 3, -1, 1]",
      "after": "derived = [1, 1, 0, 0/1]",
      "relation": "implies",
      "rule_id": "Moment_GIVEN_signed_polynomial_coefficient_moment_orientation",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依明示有號多項式矩c_k(b^(k+1)−a^(k+1))/(k+1)精確算。a>b合法，符號不取絕對值；這是受限矩模型，不把算術核對當一般FTC／可積性證書。",
        "en": "Use the DECLARED signed polynomial moment c_k(b^(k+1)−a^(k+1))/(k+1)exactly.Reversed endpoints are legal;do not take absolute values.This restricted moment model is not general FTC/integrability Evidence."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。",
          "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S050",
      "uses": [
        "G001",
        "S049"
      ],
      "before": "Polynomial moment ORIGINAL sum through coefficient3 := sum(-2/3, 0)",
      "after": "Polynomial moment ORIGINAL sum through coefficient3 = -2/3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "保留原操作數與次序作精確整數／有理運算。RAW界依bank；衍生量不重套SAFE，不用浮點近似，不把BASE整數另加legacy界。",
        "en": "Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Polynomial moment ORIGINAL sum through coefficient3，不以小數近似替代。",
          "en": "Recompute Polynomial moment ORIGINAL sum through coefficient3 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-2/3",
          "0"
        ],
        "output": "-2/3",
        "label": "Polynomial moment ORIGINAL sum through coefficient3"
      }
    },
    {
      "id": "S051",
      "uses": [
        "G001",
        "S050"
      ],
      "before": "source roles = [[0, 13, -1, 1], -1, 1]",
      "after": "derived = [-2/3]",
      "relation": "implies",
      "rule_id": "Moment_POLYNOMIAL_complete_coefficientwise_signed_sum",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整逐係數矩相加，不由少數點猜多項式／積分；正負與反向端點都保留，最簡正den且保/1。",
        "en": "Sum the COMPLETE coefficientwise moments,not guesses from sampled points.Retain signs and oriented endpoints;reduce to a positive denominator and retain/1."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。",
          "en": "Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "result": "-2/3"
  },
  "conclusion": {
    "canonical_value": "-2/3",
    "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": {
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    "question_binding": {
      "batch_id": "batch-20261007-advance-20261008-r07-definite-integral-foundations",
      "batch_path": "public/data/questions/batches/batch-20261007-advance-20261008-r07-definite-integral-foundations/questions.jsonl",
      "batch_sha256": "f08b46f392acfe2749b49fcea9773a4b95227d50a72cafa6f06b32903b00deef",
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      "original_line_sha256": "d37c0a7d0bd2226cef7818b42e34c5b2692d2a0e40da2e5350576b6662c2216a",
      "question_compact_sha256": "6cf64cadfd90d7591e82ea125c9cd7c00bc72358b8d0616419963a74c417f1bf",
      "english_row_compact_sha256": "755a74e7ed74b9525efa8f4c3ee59da48ab8b902d8a0e57f9a358e81a76490b6",
      "english_manifest_path": "public/data/i18n/editions/batch-20261007-advance-20261008-r07-definite-integral-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261007-advance-20261008-r07-definite-integral-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-definite-integral": {
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        "english_sha256": "cd470242748859a6d2bf765f69c47fa6a116eca1f2e64fb1a5edea6fe686d6a4",
        "original_path": "public/data/mko/definite-integral.json",
        "english_path": "public/data/i18n/editions/mko-definite-integral.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "528186aa825c3054db15f4ce0a03ad58f72be7e7eb5bb236c4cc7ef34ff5a470",
    "original_candidate_sha256": "3e0725feea72bbc65bdd3e8a86f96473978f866d227c10b47c9b31d2bb51dc77",
    "original_candidate_line_sha256": "3e0725feea72bbc65bdd3e8a86f96473978f866d227c10b47c9b31d2bb51dc77"
  },
  "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-20261026-011-p65",
    "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": "8b9ea3ddfab468e9e3f0a0fc0f7682987d5e030be0d7c8860fd85df19cbe183b",
      "question_first_witness_sha256": "066bfeab8fc279706861faade71b9a18a61a25235fc48700092f0488d2a6da6a"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 51,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "964db2e303e4740b32a32c49a06957060437373203258f7119194c19fa619dd9",
      "arithmetic_steps": 36
    },
    "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."
  }
}