{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261015-advance-20261113-r06-gradient-vector-0040",
  "pilot_number": "R0047-24440",
  "number": 24440,
  "family_id": "qf-mod44-gradient-rational-point",
  "title": {
    "zh-Hant": "第 24440 題：梯度向量",
    "en": "Question 24440: Evaluate Every Coordinate Derivative at the Same Exact Rational Point"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261015-advance-20261113-r06-gradient-vector-0040",
    "batch_id": "batch-20261015-advance-20261113-r06-gradient-vector-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-gradient-vector"
    ],
    "title_zh": "梯度向量",
    "stem_zh": "standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-rational-point；資料={\"dimension\":2,\"terms\":[[15,[2,1]],[16,[0,1]],[1,[0,0]]],\"point\":[[15,1],[16,2]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(2, [[1, [0, 0]], [16, [0, 1]], [15, [2, 1]]], [15, 8], [[[30, [1, 1]]], [[16, [0, 0]], [15, [2, 0]]]], [3600, 3391], exact_rational_coordinate_evaluation)",
      "explanation_zh": "Wave6第七層核對=(2, [[1, [0, 0]], [16, [0, 1]], [15, [2, 1]]], [15, 8], [[[30, [1, 1]]], [[16, [0, 0]], [15, [2, 0]]]], [3600, 3391], exact_rational_coordinate_evaluation)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 2,
      "skills": [
        "梯度向量"
      ]
    },
    "variant": {
      "family_id": "qf-mod44-gradient-rational-point",
      "seed": 202611136040,
      "parameters": {
        "dimension": 2,
        "terms": [
          [
            15,
            [
              2,
              1
            ]
          ],
          [
            16,
            [
              0,
              1
            ]
          ],
          [
            1,
            [
              0,
              0
            ]
          ]
        ],
        "point": [
          [
            15,
            1
          ],
          [
            16,
            2
          ]
        ],
        "domain": "Exact_polynomial_gradient_vector",
        "operation": "gradient-rational-point",
        "semantic_template": "qf-mod44-gradient-rational-point/{\"dimension\":2,\"point\":[[15,1],[8,1]],\"terms\":[[[1,1],[0,0]],[[15,1],[2,1]],[[16,1],[0,1]]]}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_finite_or_symbolic_model",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、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-20261113-r06-gradient-vector-0040",
    "batch_id": "batch-20261015-advance-20261113-r06-gradient-vector-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "4e7284c1a14ec6642c95488f0fa340855e2b368a25f3f5b79cc58b7791e589da",
      "family_id": "qf-mod44-gradient-rational-point",
      "seed": 202611136040,
      "semantic_template": "qf-mod44-gradient-rational-point/{\"dimension\":2,\"point\":[[15,1],[8,1]],\"terms\":[[[1,1],[0,0]],[[15,1],[2,1]],[[16,1],[0,1]]]}"
    },
    "fields": {
      "title": "Evaluate Every Coordinate Derivative at the Same Exact Rational Point",
      "stem": "Use dimension=2, raw polynomial terms=[[15, [2, 1]], [16, [0, 1]], [1, [0, 0]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Point=[15/1, 16/2]. Give (dimension, canonical_terms, point, coordinate_derivative_polynomials, gradient_values, certificate).",
      "answer": {
        "kind": "expression",
        "value": "(2, [[1, [0, 0]], [16, [0, 1]], [15, [2, 1]]], [15, 8], [[[30, [1, 1]]], [[16, [0, 0]], [15, [2, 0]]]], [3600, 3391], exact_rational_coordinate_evaluation)",
        "explanation": "Evaluate each symbolic partial at this same point with exact rational arithmetic. Keep the component order and dimension; do not replace the ordered Euclidean gradient by a sampled directional derivative. The exact result, retaining the requested order, is (2, [[1, [0, 0]], [16, [0, 1]], [15, [2, 1]]], [15, 8], [[[30, [1, 1]]], [[16, [0, 0]], [15, [2, 0]]]], [3600, 3391], exact_rational_coordinate_evaluation)."
      },
      "skills": [
        "Evaluate Every Coordinate Derivative at the Same Exact Rational Point"
      ],
      "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": "standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-rational-point；資料={\"dimension\":2,\"terms\":[[15,[2,1]],[16,[0,1]],[1,[0,0]]],\"point\":[[15,1],[16,2]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
        "en": "Use dimension=2, raw polynomial terms=[[15, [2, 1]], [16, [0, 1]], [1, [0, 0]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Point=[15/1, 16/2]. Give (dimension, canonical_terms, point, coordinate_derivative_polynomials, gradient_values, certificate)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source roles = [gradient-rational-point]",
      "after": "derived = [[dimension, canonical_terms, point, coordinate_derivative_polynomials, gradient_values, certificate]]",
      "relation": "implies",
      "rule_id": "source_ordered_fields_without_JS",
      "conditions": [],
      "explanation": {
        "zh-Hant": "本頁列完整來源欄序，不需JavaScript；gradient boundary依原mode選6／8／7欄，不補欄也不由答案猜mode。",
        "en": "List the source field order without JavaScript.Gradient boundary uses its original mode to choose6,8 or7 fields;no padding or answer-derived mode."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "Gradient ORIGINAL monomial coefficient sum0 := sum(0, 15)",
      "after": "Gradient ORIGINAL monomial coefficient sum0 = 15",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Gradient ORIGINAL monomial coefficient sum0，不以小數近似替代。",
          "en": "Recompute Gradient ORIGINAL monomial coefficient sum0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "15"
        ],
        "output": "15",
        "label": "Gradient ORIGINAL monomial coefficient sum0"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "source roles = [0, 2, [2, 1], 15]",
      "after": "derived = [15]",
      "relation": "implies",
      "rule_id": "Gradient_original_term_position_and_exponent_arity",
      "conditions": [],
      "explanation": {
        "zh-Hant": "同一ordered exponent才合併；零係數待合併後移除，固定Euclidean座標不重排。",
        "en": "Combine only identical ordered exponent vectors;remove zeros after combining and keep fixed Euclidean coordinate order."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "Gradient ORIGINAL monomial coefficient sum1 := sum(0, 16)",
      "after": "Gradient ORIGINAL monomial coefficient sum1 = 16",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Gradient ORIGINAL monomial coefficient sum1，不以小數近似替代。",
          "en": "Recompute Gradient ORIGINAL monomial coefficient sum1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "16"
        ],
        "output": "16",
        "label": "Gradient ORIGINAL monomial coefficient sum1"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "source roles = [1, 2, [0, 1], 16]",
      "after": "derived = [16]",
      "relation": "implies",
      "rule_id": "Gradient_original_term_position_and_exponent_arity",
      "conditions": [],
      "explanation": {
        "zh-Hant": "同一ordered exponent才合併；零係數待合併後移除，固定Euclidean座標不重排。",
        "en": "Combine only identical ordered exponent vectors;remove zeros after combining and keep fixed Euclidean coordinate order."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "Gradient ORIGINAL monomial coefficient sum2 := sum(0, 1)",
      "after": "Gradient ORIGINAL monomial coefficient sum2 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Gradient ORIGINAL monomial coefficient sum2，不以小數近似替代。",
          "en": "Recompute Gradient ORIGINAL monomial coefficient sum2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "1"
        ],
        "output": "1",
        "label": "Gradient ORIGINAL monomial coefficient sum2"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "source roles = [2, 2, [0, 0], 1]",
      "after": "derived = [1]",
      "relation": "implies",
      "rule_id": "Gradient_original_term_position_and_exponent_arity",
      "conditions": [],
      "explanation": {
        "zh-Hant": "同一ordered exponent才合併；零係數待合併後移除，固定Euclidean座標不重排。",
        "en": "Combine only identical ordered exponent vectors;remove zeros after combining and keep fixed Euclidean coordinate order."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "source roles = [[[15, [2, 1]], [16, [0, 1]], [1, [0, 0]]]]",
      "after": "derived = [2, [[1, [0, 0]], [16, [0, 1]], [15, [2, 1]]]]",
      "relation": "implies",
      "rule_id": "Gradient_canonical_nonzero_polynomial_terms_source_order",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按來源English canonical(exponents)序，不是數值tuple序；完整相消的零多項式可得空canonical清單。",
        "en": "Use source English canonical-exponent order,not numeric tuple order.Complete cancellation may yield an empty canonical list for the zero polynomial."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "source roles = [0, 0, [0, 0], 1]",
      "after": "derived = [0]",
      "relation": "implies",
      "rule_id": "Gradient_constant_in_coordinate_derivative_zero",
      "conditions": [],
      "explanation": {
        "zh-Hant": "該座標次數0，對此座標導數為0，不遺漏其他座標。",
        "en": "Exponent0 means derivative0 in this coordinate,without omitting other coordinates."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "source roles = [0, 1, [0, 1], 16]",
      "after": "derived = [0]",
      "relation": "implies",
      "rule_id": "Gradient_constant_in_coordinate_derivative_zero",
      "conditions": [],
      "explanation": {
        "zh-Hant": "該座標次數0，對此座標導數為0，不遺漏其他座標。",
        "en": "Exponent0 means derivative0 in this coordinate,without omitting other coordinates."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "Gradient axis0 term2 power-rule coefficient := mul(15, 2)",
      "after": "Gradient axis0 term2 power-rule coefficient = 30",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Gradient axis0 term2 power-rule coefficient，不以小數近似替代。",
          "en": "Recompute Gradient axis0 term2 power-rule coefficient with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "15",
          "2"
        ],
        "output": "30",
        "label": "Gradient axis0 term2 power-rule coefficient"
      }
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "Gradient axis0 derivative combine2 := sum(0, 30)",
      "after": "Gradient axis0 derivative combine2 = 30",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Gradient axis0 derivative combine2，不以小數近似替代。",
          "en": "Recompute Gradient axis0 derivative combine2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "30"
        ],
        "output": "30",
        "label": "Gradient axis0 derivative combine2"
      }
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "source roles = [0, 2, [2, 1], 15, 2]",
      "after": "derived = [[1, 1], 30]",
      "relation": "implies",
      "rule_id": "Gradient_coordinate_power_rule_ordered_exponents",
      "conditions": [],
      "explanation": {
        "zh-Hant": "偏導時其他變數固定，c*x_i^k變ck*x_i^(k−1)，只把這一座標次數減1。",
        "en": "Hold the other variables fixed:the coefficient becomes ck and only this coordinate exponent decreases by1."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "source roles = [1, 0, [0, 0], 1]",
      "after": "derived = [0]",
      "relation": "implies",
      "rule_id": "Gradient_constant_in_coordinate_derivative_zero",
      "conditions": [],
      "explanation": {
        "zh-Hant": "該座標次數0，對此座標導數為0，不遺漏其他座標。",
        "en": "Exponent0 means derivative0 in this coordinate,without omitting other coordinates."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "Gradient axis1 term1 power-rule coefficient := mul(16, 1)",
      "after": "Gradient axis1 term1 power-rule coefficient = 16",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Gradient axis1 term1 power-rule coefficient，不以小數近似替代。",
          "en": "Recompute Gradient axis1 term1 power-rule coefficient with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "16",
          "1"
        ],
        "output": "16",
        "label": "Gradient axis1 term1 power-rule coefficient"
      }
    },
    {
      "id": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "Gradient axis1 derivative combine1 := sum(0, 16)",
      "after": "Gradient axis1 derivative combine1 = 16",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Gradient axis1 derivative combine1，不以小數近似替代。",
          "en": "Recompute Gradient axis1 derivative combine1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "16"
        ],
        "output": "16",
        "label": "Gradient axis1 derivative combine1"
      }
    },
    {
      "id": "S017",
      "uses": [
        "G001",
        "S016"
      ],
      "before": "source roles = [1, 1, [0, 1], 16, 1]",
      "after": "derived = [[0, 0], 16]",
      "relation": "implies",
      "rule_id": "Gradient_coordinate_power_rule_ordered_exponents",
      "conditions": [],
      "explanation": {
        "zh-Hant": "偏導時其他變數固定，c*x_i^k變ck*x_i^(k−1)，只把這一座標次數減1。",
        "en": "Hold the other variables fixed:the coefficient becomes ck and only this coordinate exponent decreases by1."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S018",
      "uses": [
        "G001",
        "S017"
      ],
      "before": "Gradient axis1 term2 power-rule coefficient := mul(15, 1)",
      "after": "Gradient axis1 term2 power-rule coefficient = 15",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Gradient axis1 term2 power-rule coefficient，不以小數近似替代。",
          "en": "Recompute Gradient axis1 term2 power-rule coefficient with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "15",
          "1"
        ],
        "output": "15",
        "label": "Gradient axis1 term2 power-rule coefficient"
      }
    },
    {
      "id": "S019",
      "uses": [
        "G001",
        "S018"
      ],
      "before": "Gradient axis1 derivative combine2 := sum(0, 15)",
      "after": "Gradient axis1 derivative combine2 = 15",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Gradient axis1 derivative combine2，不以小數近似替代。",
          "en": "Recompute Gradient axis1 derivative combine2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "15"
        ],
        "output": "15",
        "label": "Gradient axis1 derivative combine2"
      }
    },
    {
      "id": "S020",
      "uses": [
        "G001",
        "S019"
      ],
      "before": "source roles = [1, 2, [2, 1], 15, 1]",
      "after": "derived = [[2, 0], 15]",
      "relation": "implies",
      "rule_id": "Gradient_coordinate_power_rule_ordered_exponents",
      "conditions": [],
      "explanation": {
        "zh-Hant": "偏導時其他變數固定，c*x_i^k變ck*x_i^(k−1)，只把這一座標次數減1。",
        "en": "Hold the other variables fixed:the coefficient becomes ck and only this coordinate exponent decreases by1."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S021",
      "uses": [
        "G001",
        "S020"
      ],
      "before": "source roles = [2, [[1, [0, 0]], [16, [0, 1]], [15, [2, 1]]]]",
      "after": "derived = [[[[30, [1, 1]]], [[16, [0, 0]], [15, [2, 0]]]]]",
      "relation": "implies",
      "rule_id": "Gradient_ALL_ordered_coordinate_polynomials",
      "conditions": [],
      "explanation": {
        "zh-Hant": "梯度是全部d個有序偏導多項式；多項式可微，故此Euclidean偏導向量表示微分。不是單方向取樣。",
        "en": "The gradient contains all d ordered component polynomials.Polynomial differentiability makes this Euclidean vector represent the differential,not a sampled single direction."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S022",
      "uses": [
        "G001",
        "S021"
      ],
      "before": "(15)^1",
      "after": "Gradient axis0 term0 coordinate0 power = 15",
      "relation": "implies",
      "rule_id": "integer_power",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以非負整數次有限乘積精確計算，零次為多項式中的乘法單位元。",
        "en": "Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": {
        "operation": "integer_power",
        "inputs": [
          "15",
          "1"
        ],
        "output": "15",
        "label": "Gradient axis0 term0 coordinate0 power"
      }
    },
    {
      "id": "S023",
      "uses": [
        "G001",
        "S022"
      ],
      "before": "Gradient axis0 term0 coordinate0 factor := mul(30, 15)",
      "after": "Gradient axis0 term0 coordinate0 factor = 450",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Gradient axis0 term0 coordinate0 factor，不以小數近似替代。",
          "en": "Recompute Gradient axis0 term0 coordinate0 factor with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "30",
          "15"
        ],
        "output": "450",
        "label": "Gradient axis0 term0 coordinate0 factor"
      }
    },
    {
      "id": "S024",
      "uses": [
        "G001",
        "S023"
      ],
      "before": "(8)^1",
      "after": "Gradient axis0 term0 coordinate1 power = 8",
      "relation": "implies",
      "rule_id": "integer_power",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以非負整數次有限乘積精確計算，零次為多項式中的乘法單位元。",
        "en": "Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": {
        "operation": "integer_power",
        "inputs": [
          "8",
          "1"
        ],
        "output": "8",
        "label": "Gradient axis0 term0 coordinate1 power"
      }
    },
    {
      "id": "S025",
      "uses": [
        "G001",
        "S024"
      ],
      "before": "Gradient axis0 term0 coordinate1 factor := mul(450, 8)",
      "after": "Gradient axis0 term0 coordinate1 factor = 3600",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Gradient axis0 term0 coordinate1 factor，不以小數近似替代。",
          "en": "Recompute Gradient axis0 term0 coordinate1 factor with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "450",
          "8"
        ],
        "output": "3600",
        "label": "Gradient axis0 term0 coordinate1 factor"
      }
    },
    {
      "id": "S026",
      "uses": [
        "G001",
        "S025"
      ],
      "before": "Gradient axis0 cumulative term0 := sum(0, 3600)",
      "after": "Gradient axis0 cumulative term0 = 3600",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Gradient axis0 cumulative term0，不以小數近似替代。",
          "en": "Recompute Gradient axis0 cumulative term0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "3600"
        ],
        "output": "3600",
        "label": "Gradient axis0 cumulative term0"
      }
    },
    {
      "id": "S027",
      "uses": [
        "G001",
        "S026"
      ],
      "before": "(15)^0",
      "after": "Gradient axis1 term0 coordinate0 power = 1",
      "relation": "implies",
      "rule_id": "integer_power",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以非負整數次有限乘積精確計算，零次為多項式中的乘法單位元。",
        "en": "Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": {
        "operation": "integer_power",
        "inputs": [
          "15",
          "0"
        ],
        "output": "1",
        "label": "Gradient axis1 term0 coordinate0 power"
      }
    },
    {
      "id": "S028",
      "uses": [
        "G001",
        "S027"
      ],
      "before": "Gradient axis1 term0 coordinate0 factor := mul(16, 1)",
      "after": "Gradient axis1 term0 coordinate0 factor = 16",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Gradient axis1 term0 coordinate0 factor，不以小數近似替代。",
          "en": "Recompute Gradient axis1 term0 coordinate0 factor with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "16",
          "1"
        ],
        "output": "16",
        "label": "Gradient axis1 term0 coordinate0 factor"
      }
    },
    {
      "id": "S029",
      "uses": [
        "G001",
        "S028"
      ],
      "before": "(8)^0",
      "after": "Gradient axis1 term0 coordinate1 power = 1",
      "relation": "implies",
      "rule_id": "integer_power",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以非負整數次有限乘積精確計算，零次為多項式中的乘法單位元。",
        "en": "Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": {
        "operation": "integer_power",
        "inputs": [
          "8",
          "0"
        ],
        "output": "1",
        "label": "Gradient axis1 term0 coordinate1 power"
      }
    },
    {
      "id": "S030",
      "uses": [
        "G001",
        "S029"
      ],
      "before": "Gradient axis1 term0 coordinate1 factor := mul(16, 1)",
      "after": "Gradient axis1 term0 coordinate1 factor = 16",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Gradient axis1 term0 coordinate1 factor，不以小數近似替代。",
          "en": "Recompute Gradient axis1 term0 coordinate1 factor with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "16",
          "1"
        ],
        "output": "16",
        "label": "Gradient axis1 term0 coordinate1 factor"
      }
    },
    {
      "id": "S031",
      "uses": [
        "G001",
        "S030"
      ],
      "before": "(15)^2",
      "after": "Gradient axis1 term1 coordinate0 power = 225",
      "relation": "implies",
      "rule_id": "integer_power",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以非負整數次有限乘積精確計算，零次為多項式中的乘法單位元。",
        "en": "Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": {
        "operation": "integer_power",
        "inputs": [
          "15",
          "2"
        ],
        "output": "225",
        "label": "Gradient axis1 term1 coordinate0 power"
      }
    },
    {
      "id": "S032",
      "uses": [
        "G001",
        "S031"
      ],
      "before": "Gradient axis1 term1 coordinate0 factor := mul(15, 225)",
      "after": "Gradient axis1 term1 coordinate0 factor = 3375",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Gradient axis1 term1 coordinate0 factor，不以小數近似替代。",
          "en": "Recompute Gradient axis1 term1 coordinate0 factor with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "15",
          "225"
        ],
        "output": "3375",
        "label": "Gradient axis1 term1 coordinate0 factor"
      }
    },
    {
      "id": "S033",
      "uses": [
        "G001",
        "S032"
      ],
      "before": "(8)^0",
      "after": "Gradient axis1 term1 coordinate1 power = 1",
      "relation": "implies",
      "rule_id": "integer_power",
      "conditions": [],
      "explanation": {
        "zh-Hant": "以非負整數次有限乘積精確計算，零次為多項式中的乘法單位元。",
        "en": "Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": {
        "operation": "integer_power",
        "inputs": [
          "8",
          "0"
        ],
        "output": "1",
        "label": "Gradient axis1 term1 coordinate1 power"
      }
    },
    {
      "id": "S034",
      "uses": [
        "G001",
        "S033"
      ],
      "before": "Gradient axis1 term1 coordinate1 factor := mul(3375, 1)",
      "after": "Gradient axis1 term1 coordinate1 factor = 3375",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Gradient axis1 term1 coordinate1 factor，不以小數近似替代。",
          "en": "Recompute Gradient axis1 term1 coordinate1 factor with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "3375",
          "1"
        ],
        "output": "3375",
        "label": "Gradient axis1 term1 coordinate1 factor"
      }
    },
    {
      "id": "S035",
      "uses": [
        "G001",
        "S034"
      ],
      "before": "Gradient axis1 cumulative term0 := sum(0, 16)",
      "after": "Gradient axis1 cumulative term0 = 16",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Gradient axis1 cumulative term0，不以小數近似替代。",
          "en": "Recompute Gradient axis1 cumulative term0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "0",
          "16"
        ],
        "output": "16",
        "label": "Gradient axis1 cumulative term0"
      }
    },
    {
      "id": "S036",
      "uses": [
        "G001",
        "S035"
      ],
      "before": "Gradient axis1 cumulative term1 := sum(16, 3375)",
      "after": "Gradient axis1 cumulative term1 = 3391",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。",
        "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 Gradient axis1 cumulative term1，不以小數近似替代。",
          "en": "Recompute Gradient axis1 cumulative term1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "16",
          "3375"
        ],
        "output": "3391",
        "label": "Gradient axis1 cumulative term1"
      }
    },
    {
      "id": "S037",
      "uses": [
        "G001",
        "S036"
      ],
      "before": "source roles = [2, [15, 8], [[[30, [1, 1]]], [[16, [0, 0]], [15, [2, 0]]]]]",
      "after": "derived = [[3600, 3391], 否, 否]",
      "relation": "implies",
      "rule_id": "Gradient_SAME_point_all_coordinates_stationary_not_extremum",
      "conditions": [],
      "explanation": {
        "zh-Hant": "全部分量同一point代值才判stationary；零梯度不自動給extremum證書，No亦不否定常數函數等特例真有極值。",
        "en": "Evaluate every component at the same point before deciding stationarity.Zero gradient supplies no automatic extremum certificate;No does not deny actual extrema of special cases such as constant functions."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "result": [
      2,
      [
        [
          "1",
          [
            0,
            0
          ]
        ],
        [
          "16",
          [
            0,
            1
          ]
        ],
        [
          "15",
          [
            2,
            1
          ]
        ]
      ],
      [
        "15",
        "8"
      ],
      [
        [
          [
            "30",
            [
              1,
              1
            ]
          ]
        ],
        [
          [
            "16",
            [
              0,
              0
            ]
          ],
          [
            "15",
            [
              2,
              0
            ]
          ]
        ]
      ],
      [
        "3600",
        "3391"
      ],
      "exact_rational_coordinate_evaluation"
    ]
  },
  "conclusion": {
    "canonical_value": "(2, [[1, [0, 0]], [16, [0, 1]], [15, [2, 1]]], [15, 8], [[[30, [1, 1]]], [[16, [0, 0]], [15, [2, 0]]]], [3600, 3391], exact_rational_coordinate_evaluation)",
    "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-20261113-r06-gradient-vector-foundations",
      "batch_path": "public/data/questions/batches/batch-20261015-advance-20261113-r06-gradient-vector-foundations/questions.jsonl",
      "batch_sha256": "2126db15a137e63c9415addb2bbac655ac3a34f6aa471dc44c138067bd9af76d",
      "original_line": 40,
      "original_line_sha256": "7bde7f180d7c78745c8695467ea69af75112b3ae7d65175134a8a8dfa960e1cd",
      "question_compact_sha256": "4e7284c1a14ec6642c95488f0fa340855e2b368a25f3f5b79cc58b7791e589da",
      "english_row_compact_sha256": "9321c11aea7133f52986cad9d78158af4c01f61d1678aecd3582d2dca1ad5911",
      "english_manifest_path": "public/data/i18n/editions/batch-20261015-advance-20261113-r06-gradient-vector-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261015-advance-20261113-r06-gradient-vector-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-gradient-vector": {
        "sha256": "953e582b90145e3769b9b22891783891e5dbfe6a99446edc1b06a277683029fa",
        "english_sha256": "6ba53da50d59e50b734006c70a0c1e5cb7e4986a4a57e6d0b1f27a8ae84bd363",
        "original_path": "public/data/mko/gradient-vector.json",
        "english_path": "public/data/i18n/editions/mko-gradient-vector.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "b3aad195dad743751e4c8a4a6ce44f091c3824da1e52c21e9266d4ccd1dc1b68",
    "original_candidate_sha256": "86ed8a9bbbef09d0bb3e6bb3e0355dde645c3c11f0f41fa2f6f2fb8bdcf97b36",
    "original_candidate_line_sha256": "86ed8a9bbbef09d0bb3e6bb3e0355dde645c3c11f0f41fa2f6f2fb8bdcf97b36"
  },
  "historical_source_state": {
    "review_state": {
      "derivation": "candidate_authored_and_locally_checked",
      "source_question_state": "candidate",
      "independent_AI_or_human_review": "NOT_PERFORMED",
      "formalization": "not_formalized",
      "Lean": "NOT_RUN"
    },
    "scope_note": {
      "zh-Hant": "來源綁定教學推導候選；精確核對不等於獨立AI／人工審查或一般形式證明。原題、答案、MKO、公式、AST與歷史不變。",
      "en": "Source-bound pedagogical derivation candidate.Exact checking is not independent AI/human review or general formal proof.Original questions,answers,MKOs,formulas,ASTs and history remain unchanged."
    },
    "label": "Frozen producer metadata; current per-question review is separate below."
  },
  "revision": {
    "id": "review-20261026-009-p48",
    "review_date": "2026-10-26",
    "owner_authorized_reader_publication": true,
    "scope": "One hundred sixty family prototypes and their mathematical/bilingual reasoning variants, stems, boundary statements and every distinct explanation shape were reviewed in a context independent of the producer; parameter substitutions, source bindings, full math models, both answer strings and every arithmetic step were independently reconstructed per question from the original parameters only, with negative controls. This grouped method does not claim a separate literal AI reading of every repeated sentence or an extra AI voter.",
    "review_receipts": {
      "semantic_review_sha256": "9b43b9823e9bd2478d6caa5b57ffed161c2c8956af86974e901d5662ad994386",
      "question_first_witness_sha256": "eb97596068e2d5648033bab2b8f6ca9b653bbe8cb77c0f8e1d16d0c40e46f50b"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 37,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "a89000fd83218932387fd45c7c67a4bd0040f19221cfccd22535c3571ba584df",
      "arithmetic_steps": 24
    },
    "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."
  }
}