{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261015-advance-20261118-r05-p-adic-valuation-integers-0039",
  "pilot_number": "R0053-27464",
  "number": 27464,
  "family_id": "qf-mod49-pval-product-additivity",
  "title": {
    "zh-Hant": "第 27464 題：整數p進賦值",
    "en": "Question 27464: Integer Valuations Add under Products, Including the Fixed Zero Convention"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261015-advance-20261118-r05-p-adic-valuation-integers-0039",
    "batch_id": "batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-p-adic-valuation-integers"
    ],
    "title_zh": "整數p進賦值",
    "stem_zh": "prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-product-additivity；資料={\"p\":3,\"a\":42,\"b\":-135}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(3, 42, -135, 1, 3, -5670, 4, 4, 是, infinity_plus_k_equals_infinity)",
      "explanation_zh": "Wave6第十二層核對=(3, 42, -135, 1, 3, -5670, 4, 4, 是, infinity_plus_k_equals_infinity)；依完整明示資料獨立重算；negative integers合法，product additivity與sum inequality exact重算；sum equality只在valuations unequal時保證；composite base不是p-adic valuation。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 2,
      "skills": [
        "整數p進賦值"
      ]
    },
    "variant": {
      "family_id": "qf-mod49-pval-product-additivity",
      "seed": 202611185039,
      "parameters": {
        "p": 3,
        "a": 42,
        "b": -135,
        "domain": "Exact_integer_p_adic_valuation",
        "operation": "pval-product-additivity",
        "semantic_template": "pval-product-additivity|{a:42,b:-135,p:3}"
      }
    },
    "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；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、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-20261118-r05-p-adic-valuation-integers-0039",
    "batch_id": "batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "8cbdfd4990e28be437250cca436ce3397c5911041df9d481d7da3dd57d6c5b1c",
      "family_id": "qf-mod49-pval-product-additivity",
      "seed": 202611185039,
      "semantic_template": "pval-product-additivity|{a:42,b:-135,p:3}"
    },
    "fields": {
      "title": "Integer Valuations Add under Products, Including the Fixed Zero Convention",
      "stem": "Use prime p=3, integers a=42, b=-135, with v_p(0)=infinity. Give (p, a, b, valuation_a, valuation_b, product, valuation_product, sum_of_valuations, additivity, infinity_convention).",
      "answer": {
        "kind": "expression",
        "value": "(3, 42, -135, 1, 3, -5670, 4, 4, Yes, infinity_plus_k_equals_infinity)",
        "explanation": "Compute both valuations and the product valuation. If either factor is zero, the valuation sum is the single infinity sentinel. Otherwise add the finite exponents. This integer instance is not a construction of a p-adic completion. The exact result, retaining the requested order, is (3, 42, -135, 1, 3, -5670, 4, 4, Yes, infinity_plus_k_equals_infinity)."
      },
      "skills": [
        "Integer Valuations Add under Products, Including the Fixed Zero Convention"
      ],
      "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": "prime p、absolute-value factor extraction與v_p(0)=infinity責任。操作=pval-product-additivity；資料={\"p\":3,\"a\":42,\"b\":-135}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
        "en": "Use prime p=3, integers a=42, b=-135, with v_p(0)=infinity. Give (p, a, b, valuation_a, valuation_b, product, valuation_product, sum_of_valuations, additivity, infinity_convention)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source roles = [pval-product-additivity]",
      "after": "derived = [[p, a, b, valuation_a, valuation_b, product, valuation_product, sum_of_valuations, additivity, infinity_convention]]",
      "relation": "implies",
      "rule_id": "source_ordered_fields_without_JS",
      "conditions": [],
      "explanation": {
        "zh-Hant": "此頁直接列完整來源欄序與mode特定角色，不靠JavaScript或由答案外形猜介面；No、undefined、infinity各保原意。",
        "en": "List the complete source field order and mode-specific roles without JavaScript or answer-shape inference;No,undefined and infinity retain their distinct meanings."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。",
          "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "Valuation ORIGINAL a ORIGINAL division step1 := div(42, 3)",
      "after": "Valuation ORIGINAL a ORIGINAL division step1 = 14",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Valuation ORIGINAL a ORIGINAL division step1，不以小數近似替代。",
          "en": "Recompute Valuation ORIGINAL a ORIGINAL division step1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "42",
          "3"
        ],
        "output": "14",
        "label": "Valuation ORIGINAL a ORIGINAL division step1"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "Valuation ORIGINAL a dividing prime power1 := mul(1, 3)",
      "after": "Valuation ORIGINAL a dividing prime power1 = 3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Valuation ORIGINAL a dividing prime power1，不以小數近似替代。",
          "en": "Recompute Valuation ORIGINAL a dividing prime power1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "3"
        ],
        "output": "3",
        "label": "Valuation ORIGINAL a dividing prime power1"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "source roles = [Valuation ORIGINAL a, 3, 42, 42, 1]",
      "after": "derived = [14, 3, 0, 是]",
      "relation": "implies",
      "rule_id": "Valuation_EACH_division_source_remainder_and_factor_invariant",
      "conditions": [],
      "explanation": {
        "zh-Hant": "當前unit可整除p才除；每步|a|=p^k*unit不變，且正unit嚴格下降，所以有限停止。這些衍生整數不套raw-input的1e6上限。",
        "en": "Divide only when the current unit is divisible by p.Each step preserves |a|=p^k*unit and strictly decreases the positive unit,so it terminates.Derived integers do not inherit the raw-input1e6 cap."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。",
          "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "Valuation ORIGINAL a terminal factorization product := mul(3, 14)",
      "after": "Valuation ORIGINAL a terminal factorization product = 42",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Valuation ORIGINAL a terminal factorization product，不以小數近似替代。",
          "en": "Recompute Valuation ORIGINAL a terminal factorization product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "3",
          "14"
        ],
        "output": "42",
        "label": "Valuation ORIGINAL a terminal factorization product"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "source roles = [Valuation ORIGINAL a, 3, 42, 42, 1, 14, [3]]",
      "after": "derived = [42, 2, 是, 是]",
      "relation": "implies",
      "rule_id": "Valuation_terminal_NONdivisible_unit_highest_power",
      "conditions": [],
      "explanation": {
        "zh-Hant": "停止時p不整除unit，所以p^(k+1)不整除|a|，k就是最高指數。負號不改整除性，factorization回到|a|，不是把負a等於正product。",
        "en": "At termination p does not divide the unit,so p^(k+1) does not divide |a| and k is maximal.A sign does not affect divisibility;the factorization is for |a|,not negative a equalling a positive product."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。",
          "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "Valuation ORIGINAL b ORIGINAL division step1 := div(135, 3)",
      "after": "Valuation ORIGINAL b ORIGINAL division step1 = 45",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Valuation ORIGINAL b ORIGINAL division step1，不以小數近似替代。",
          "en": "Recompute Valuation ORIGINAL b ORIGINAL division step1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "135",
          "3"
        ],
        "output": "45",
        "label": "Valuation ORIGINAL b ORIGINAL division step1"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "Valuation ORIGINAL b dividing prime power1 := mul(1, 3)",
      "after": "Valuation ORIGINAL b dividing prime power1 = 3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Valuation ORIGINAL b dividing prime power1，不以小數近似替代。",
          "en": "Recompute Valuation ORIGINAL b dividing prime power1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "3"
        ],
        "output": "3",
        "label": "Valuation ORIGINAL b dividing prime power1"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "source roles = [Valuation ORIGINAL b, 3, 135, 135, 1]",
      "after": "derived = [45, 3, 0, 是]",
      "relation": "implies",
      "rule_id": "Valuation_EACH_division_source_remainder_and_factor_invariant",
      "conditions": [],
      "explanation": {
        "zh-Hant": "當前unit可整除p才除；每步|a|=p^k*unit不變，且正unit嚴格下降，所以有限停止。這些衍生整數不套raw-input的1e6上限。",
        "en": "Divide only when the current unit is divisible by p.Each step preserves |a|=p^k*unit and strictly decreases the positive unit,so it terminates.Derived integers do not inherit the raw-input1e6 cap."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。",
          "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "Valuation ORIGINAL b ORIGINAL division step2 := div(45, 3)",
      "after": "Valuation ORIGINAL b ORIGINAL division step2 = 15",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Valuation ORIGINAL b ORIGINAL division step2，不以小數近似替代。",
          "en": "Recompute Valuation ORIGINAL b ORIGINAL division step2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "45",
          "3"
        ],
        "output": "15",
        "label": "Valuation ORIGINAL b ORIGINAL division step2"
      }
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "Valuation ORIGINAL b dividing prime power2 := mul(3, 3)",
      "after": "Valuation ORIGINAL b dividing prime power2 = 9",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Valuation ORIGINAL b dividing prime power2，不以小數近似替代。",
          "en": "Recompute Valuation ORIGINAL b dividing prime power2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "3",
          "3"
        ],
        "output": "9",
        "label": "Valuation ORIGINAL b dividing prime power2"
      }
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "source roles = [Valuation ORIGINAL b, 3, 135, 45, 2]",
      "after": "derived = [15, 9, 0, 是]",
      "relation": "implies",
      "rule_id": "Valuation_EACH_division_source_remainder_and_factor_invariant",
      "conditions": [],
      "explanation": {
        "zh-Hant": "當前unit可整除p才除；每步|a|=p^k*unit不變，且正unit嚴格下降，所以有限停止。這些衍生整數不套raw-input的1e6上限。",
        "en": "Divide only when the current unit is divisible by p.Each step preserves |a|=p^k*unit and strictly decreases the positive unit,so it terminates.Derived integers do not inherit the raw-input1e6 cap."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。",
          "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "Valuation ORIGINAL b ORIGINAL division step3 := div(15, 3)",
      "after": "Valuation ORIGINAL b ORIGINAL division step3 = 5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Valuation ORIGINAL b ORIGINAL division step3，不以小數近似替代。",
          "en": "Recompute Valuation ORIGINAL b ORIGINAL division step3 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "15",
          "3"
        ],
        "output": "5",
        "label": "Valuation ORIGINAL b ORIGINAL division step3"
      }
    },
    {
      "id": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "Valuation ORIGINAL b dividing prime power3 := mul(9, 3)",
      "after": "Valuation ORIGINAL b dividing prime power3 = 27",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Valuation ORIGINAL b dividing prime power3，不以小數近似替代。",
          "en": "Recompute Valuation ORIGINAL b dividing prime power3 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "9",
          "3"
        ],
        "output": "27",
        "label": "Valuation ORIGINAL b dividing prime power3"
      }
    },
    {
      "id": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "source roles = [Valuation ORIGINAL b, 3, 135, 15, 3]",
      "after": "derived = [5, 27, 0, 是]",
      "relation": "implies",
      "rule_id": "Valuation_EACH_division_source_remainder_and_factor_invariant",
      "conditions": [],
      "explanation": {
        "zh-Hant": "當前unit可整除p才除；每步|a|=p^k*unit不變，且正unit嚴格下降，所以有限停止。這些衍生整數不套raw-input的1e6上限。",
        "en": "Divide only when the current unit is divisible by p.Each step preserves |a|=p^k*unit and strictly decreases the positive unit,so it terminates.Derived integers do not inherit the raw-input1e6 cap."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。",
          "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "Valuation ORIGINAL b terminal factorization product := mul(27, 5)",
      "after": "Valuation ORIGINAL b terminal factorization product = 135",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Valuation ORIGINAL b terminal factorization product，不以小數近似替代。",
          "en": "Recompute Valuation ORIGINAL b terminal factorization product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "27",
          "5"
        ],
        "output": "135",
        "label": "Valuation ORIGINAL b terminal factorization product"
      }
    },
    {
      "id": "S017",
      "uses": [
        "G001",
        "S016"
      ],
      "before": "source roles = [Valuation ORIGINAL b, 3, -135, 135, 3, 5, [3, 9, 27]]",
      "after": "derived = [135, 2, 是, 是]",
      "relation": "implies",
      "rule_id": "Valuation_terminal_NONdivisible_unit_highest_power",
      "conditions": [],
      "explanation": {
        "zh-Hant": "停止時p不整除unit，所以p^(k+1)不整除|a|，k就是最高指數。負號不改整除性，factorization回到|a|，不是把負a等於正product。",
        "en": "At termination p does not divide the unit,so p^(k+1) does not divide |a| and k is maximal.A sign does not affect divisibility;the factorization is for |a|,not negative a equalling a positive product."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。",
          "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S018",
      "uses": [
        "G001",
        "S017"
      ],
      "before": "Valuation ORIGINAL a times b := mul(42, -135)",
      "after": "Valuation ORIGINAL a times b = -5670",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Valuation ORIGINAL a times b，不以小數近似替代。",
          "en": "Recompute Valuation ORIGINAL a times b with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "42",
          "-135"
        ],
        "output": "-5670",
        "label": "Valuation ORIGINAL a times b"
      }
    },
    {
      "id": "S019",
      "uses": [
        "G001",
        "S018"
      ],
      "before": "Valuation ACTUAL product ORIGINAL division step1 := div(5670, 3)",
      "after": "Valuation ACTUAL product ORIGINAL division step1 = 1890",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Valuation ACTUAL product ORIGINAL division step1，不以小數近似替代。",
          "en": "Recompute Valuation ACTUAL product ORIGINAL division step1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "5670",
          "3"
        ],
        "output": "1890",
        "label": "Valuation ACTUAL product ORIGINAL division step1"
      }
    },
    {
      "id": "S020",
      "uses": [
        "G001",
        "S019"
      ],
      "before": "Valuation ACTUAL product dividing prime power1 := mul(1, 3)",
      "after": "Valuation ACTUAL product dividing prime power1 = 3",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Valuation ACTUAL product dividing prime power1，不以小數近似替代。",
          "en": "Recompute Valuation ACTUAL product dividing prime power1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "3"
        ],
        "output": "3",
        "label": "Valuation ACTUAL product dividing prime power1"
      }
    },
    {
      "id": "S021",
      "uses": [
        "G001",
        "S020"
      ],
      "before": "source roles = [Valuation ACTUAL product, 3, 5670, 5670, 1]",
      "after": "derived = [1890, 3, 0, 是]",
      "relation": "implies",
      "rule_id": "Valuation_EACH_division_source_remainder_and_factor_invariant",
      "conditions": [],
      "explanation": {
        "zh-Hant": "當前unit可整除p才除；每步|a|=p^k*unit不變，且正unit嚴格下降，所以有限停止。這些衍生整數不套raw-input的1e6上限。",
        "en": "Divide only when the current unit is divisible by p.Each step preserves |a|=p^k*unit and strictly decreases the positive unit,so it terminates.Derived integers do not inherit the raw-input1e6 cap."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。",
          "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S022",
      "uses": [
        "G001",
        "S021"
      ],
      "before": "Valuation ACTUAL product ORIGINAL division step2 := div(1890, 3)",
      "after": "Valuation ACTUAL product ORIGINAL division step2 = 630",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Valuation ACTUAL product ORIGINAL division step2，不以小數近似替代。",
          "en": "Recompute Valuation ACTUAL product ORIGINAL division step2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "1890",
          "3"
        ],
        "output": "630",
        "label": "Valuation ACTUAL product ORIGINAL division step2"
      }
    },
    {
      "id": "S023",
      "uses": [
        "G001",
        "S022"
      ],
      "before": "Valuation ACTUAL product dividing prime power2 := mul(3, 3)",
      "after": "Valuation ACTUAL product dividing prime power2 = 9",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Valuation ACTUAL product dividing prime power2，不以小數近似替代。",
          "en": "Recompute Valuation ACTUAL product dividing prime power2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "3",
          "3"
        ],
        "output": "9",
        "label": "Valuation ACTUAL product dividing prime power2"
      }
    },
    {
      "id": "S024",
      "uses": [
        "G001",
        "S023"
      ],
      "before": "source roles = [Valuation ACTUAL product, 3, 5670, 1890, 2]",
      "after": "derived = [630, 9, 0, 是]",
      "relation": "implies",
      "rule_id": "Valuation_EACH_division_source_remainder_and_factor_invariant",
      "conditions": [],
      "explanation": {
        "zh-Hant": "當前unit可整除p才除；每步|a|=p^k*unit不變，且正unit嚴格下降，所以有限停止。這些衍生整數不套raw-input的1e6上限。",
        "en": "Divide only when the current unit is divisible by p.Each step preserves |a|=p^k*unit and strictly decreases the positive unit,so it terminates.Derived integers do not inherit the raw-input1e6 cap."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。",
          "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S025",
      "uses": [
        "G001",
        "S024"
      ],
      "before": "Valuation ACTUAL product ORIGINAL division step3 := div(630, 3)",
      "after": "Valuation ACTUAL product ORIGINAL division step3 = 210",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Valuation ACTUAL product ORIGINAL division step3，不以小數近似替代。",
          "en": "Recompute Valuation ACTUAL product ORIGINAL division step3 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "630",
          "3"
        ],
        "output": "210",
        "label": "Valuation ACTUAL product ORIGINAL division step3"
      }
    },
    {
      "id": "S026",
      "uses": [
        "G001",
        "S025"
      ],
      "before": "Valuation ACTUAL product dividing prime power3 := mul(9, 3)",
      "after": "Valuation ACTUAL product dividing prime power3 = 27",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Valuation ACTUAL product dividing prime power3，不以小數近似替代。",
          "en": "Recompute Valuation ACTUAL product dividing prime power3 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "9",
          "3"
        ],
        "output": "27",
        "label": "Valuation ACTUAL product dividing prime power3"
      }
    },
    {
      "id": "S027",
      "uses": [
        "G001",
        "S026"
      ],
      "before": "source roles = [Valuation ACTUAL product, 3, 5670, 630, 3]",
      "after": "derived = [210, 27, 0, 是]",
      "relation": "implies",
      "rule_id": "Valuation_EACH_division_source_remainder_and_factor_invariant",
      "conditions": [],
      "explanation": {
        "zh-Hant": "當前unit可整除p才除；每步|a|=p^k*unit不變，且正unit嚴格下降，所以有限停止。這些衍生整數不套raw-input的1e6上限。",
        "en": "Divide only when the current unit is divisible by p.Each step preserves |a|=p^k*unit and strictly decreases the positive unit,so it terminates.Derived integers do not inherit the raw-input1e6 cap."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。",
          "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S028",
      "uses": [
        "G001",
        "S027"
      ],
      "before": "Valuation ACTUAL product ORIGINAL division step4 := div(210, 3)",
      "after": "Valuation ACTUAL product ORIGINAL division step4 = 70",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Valuation ACTUAL product ORIGINAL division step4，不以小數近似替代。",
          "en": "Recompute Valuation ACTUAL product ORIGINAL division step4 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "div",
        "inputs": [
          "210",
          "3"
        ],
        "output": "70",
        "label": "Valuation ACTUAL product ORIGINAL division step4"
      }
    },
    {
      "id": "S029",
      "uses": [
        "G001",
        "S028"
      ],
      "before": "Valuation ACTUAL product dividing prime power4 := mul(27, 3)",
      "after": "Valuation ACTUAL product dividing prime power4 = 81",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Valuation ACTUAL product dividing prime power4，不以小數近似替代。",
          "en": "Recompute Valuation ACTUAL product dividing prime power4 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "27",
          "3"
        ],
        "output": "81",
        "label": "Valuation ACTUAL product dividing prime power4"
      }
    },
    {
      "id": "S030",
      "uses": [
        "G001",
        "S029"
      ],
      "before": "source roles = [Valuation ACTUAL product, 3, 5670, 210, 4]",
      "after": "derived = [70, 81, 0, 是]",
      "relation": "implies",
      "rule_id": "Valuation_EACH_division_source_remainder_and_factor_invariant",
      "conditions": [],
      "explanation": {
        "zh-Hant": "當前unit可整除p才除；每步|a|=p^k*unit不變，且正unit嚴格下降，所以有限停止。這些衍生整數不套raw-input的1e6上限。",
        "en": "Divide only when the current unit is divisible by p.Each step preserves |a|=p^k*unit and strictly decreases the positive unit,so it terminates.Derived integers do not inherit the raw-input1e6 cap."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。",
          "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S031",
      "uses": [
        "G001",
        "S030"
      ],
      "before": "Valuation ACTUAL product terminal factorization product := mul(81, 70)",
      "after": "Valuation ACTUAL product terminal factorization product = 5670",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Valuation ACTUAL product terminal factorization product，不以小數近似替代。",
          "en": "Recompute Valuation ACTUAL product terminal factorization product with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "81",
          "70"
        ],
        "output": "5670",
        "label": "Valuation ACTUAL product terminal factorization product"
      }
    },
    {
      "id": "S032",
      "uses": [
        "G001",
        "S031"
      ],
      "before": "source roles = [Valuation ACTUAL product, 3, -5670, 5670, 4, 70, [3, 9, 27, 81]]",
      "after": "derived = [5670, 1, 是, 是]",
      "relation": "implies",
      "rule_id": "Valuation_terminal_NONdivisible_unit_highest_power",
      "conditions": [],
      "explanation": {
        "zh-Hant": "停止時p不整除unit，所以p^(k+1)不整除|a|，k就是最高指數。負號不改整除性，factorization回到|a|，不是把負a等於正product。",
        "en": "At termination p does not divide the unit,so p^(k+1) does not divide |a| and k is maximal.A sign does not affect divisibility;the factorization is for |a|,not negative a equalling a positive product."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。",
          "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S033",
      "uses": [
        "G001",
        "S032"
      ],
      "before": "Valuation ORIGINAL exponent addition := sum(1, 3)",
      "after": "Valuation ORIGINAL exponent addition = 4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。",
        "en": "Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Valuation ORIGINAL exponent addition，不以小數近似替代。",
          "en": "Recompute Valuation ORIGINAL exponent addition with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "1",
          "3"
        ],
        "output": "4",
        "label": "Valuation ORIGINAL exponent addition"
      }
    },
    {
      "id": "S034",
      "uses": [
        "G001",
        "S033"
      ],
      "before": "source roles = [3, 42, -135, 1, 3, 14, 5]",
      "after": "derived = [-5670, 4, 4, 是]",
      "relation": "implies",
      "rule_id": "Valuation_GENERAL_product_primeness_unit_product_or_zero",
      "conditions": [],
      "explanation": {
        "zh-Hant": "非零時a=±p^r*u,b=±p^s*v，prime p不整除u,v就不整除uv（Euclid lemma），故最高指數恰r+s。若一因子0則product0，依同一infinity約定相加；derived product可達1e12，仍是SAFE，不套raw1e6。",
        "en": "For nonzero factors write a=±p^r*u,b=±p^s*v.Primality and Euclid lemma keep uv indivisible by p,so the exact exponent is r+s.If either factor is0,the product is0 and the same infinity convention applies.A derived product may reach1e12 while remaining SAFE;do not reapply the raw1e6 bound."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。",
          "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "result": [
      3,
      42,
      -135,
      1,
      3,
      -5670,
      4,
      4,
      true,
      "infinity_plus_k_equals_infinity"
    ]
  },
  "conclusion": {
    "canonical_value": "(3, 42, -135, 1, 3, -5670, 4, 4, 是, infinity_plus_k_equals_infinity)",
    "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-20261118-r05-p-adic-valuation-integers-foundations",
      "batch_path": "public/data/questions/batches/batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations/questions.jsonl",
      "batch_sha256": "a2f6d43cd783e7dd6234ba65b2293735a102a99a2e2b7cb02a64e5f2431ede08",
      "original_line": 39,
      "original_line_sha256": "958461f1aeeddf78bdc74bc40c135244705e2cbf6fe6f7f91a880ce3bda1d95e",
      "question_compact_sha256": "8cbdfd4990e28be437250cca436ce3397c5911041df9d481d7da3dd57d6c5b1c",
      "english_row_compact_sha256": "689db2deb739835b9ef4e46d8fa0f36fde6dd81393cf19eb050f01afd3dff43c",
      "english_manifest_path": "public/data/i18n/editions/batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-p-adic-valuation-integers": {
        "sha256": "9805aa95014a92e3ee82b127895171c719f8deb0548a932c13fb85c07d782a40",
        "english_sha256": "13f277e55f58c0bbb023895d24c591527ec1f4bd6f223d42e1a8f1070ec3facc",
        "original_path": "public/data/mko/p-adic-valuation-integers.json",
        "english_path": "public/data/i18n/editions/mko-p-adic-valuation-integers.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "a54409c0bc3f68730e1072c6d3d772970462c1bc7ffc203b4030a6136a610283",
    "original_candidate_sha256": "40b23b486a6b8857d583127ba03673dfbff6109fa0f6c1b77badce52f79ffbfa",
    "original_candidate_line_sha256": "40b23b486a6b8857d583127ba03673dfbff6109fa0f6c1b77badce52f79ffbfa"
  },
  "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-p54",
    "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": "1b7a62514347b82518305d5feb4369a52d7d44c442a0f2b131e9eb3cd7b45f62",
      "question_first_witness_sha256": "79dc2a1077081f3ea2a03f75e32778163bf05e814cdf4c29ed86e567295f9ee1"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 34,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "6c932995f0cb4f6a89d6badd25d8b35e1a6cbc67bc431544344ee1b62f476cef",
      "arithmetic_steps": 21
    },
    "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."
  }
}