{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261013-advance-20261023-r07-multivariable-real-function-0073",
  "pilot_number": "R0020-10698",
  "number": 10698,
  "family_id": "qf-mod23-multivar-x-slice-table",
  "title": {
    "zh-Hant": "第 10698 題：二變數實函數",
    "en": "Question 10698: A Fixed-Second-Coordinate Slice on Listed Inputs Only"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261013-advance-20261023-r07-multivariable-real-function-0073",
    "batch_id": "batch-20261013-advance-20261023-r07-multivariable-real-function-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-multivariable-real-function"
    ],
    "title_zh": "二變數實函數",
    "stem_zh": "明示R2到R多項式與座標角色。操作=multivar-x-slice-table；資料={\"coefficients\":[10,0,-2,1,0,-1],\"y\":0,\"xs\":[-1,0,-2]}。請依 [coefficients,fixed_y,xs,table] 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "([10, 0, -2, 1, 0, -1], 0, [-2, -1, 0], [[-2, 14], [-1, 11], [0, 10]])",
      "explanation_zh": "現代橋梁核對=([10, 0, -2, 1, 0, -1], 0, [-2, -1, 0], [[-2, 14], [-1, 11], [0, 10]])；依完整明示資料重算；完整明示的有限／精確切片不自動升格為一般無限結論；未聲明全域連續或可微；不外推向量值函數。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 3,
      "skills": [
        "二變數實函數"
      ]
    },
    "variant": {
      "family_id": "qf-mod23-multivar-x-slice-table",
      "seed": 202610237073,
      "parameters": {
        "coefficients": [
          10,
          0,
          -2,
          1,
          0,
          -1
        ],
        "y": 0,
        "xs": [
          -1,
          0,
          -2
        ],
        "domain": "Polynomial_R2_to_R_slice",
        "operation": "multivar-x-slice-table",
        "semantic_template": "qf-mod23-multivar-x-slice-table/{\"coefficients\":[10,0,-2,1,0,-1],\"xs\":[-2,-1,0],\"y\":0}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_finite_model",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "產生端候選須由主端重算；不構成人工審定、一般無限定理、Markov／可微聲明、Lean執行或形式Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-13",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261013-advance-20261023-r07-multivariable-real-function-0073",
    "batch_id": "batch-20261013-advance-20261023-r07-multivariable-real-function-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "e510807f4ff5bb52a28633465a530c0baa4d2df1d3ee17cc434be323f9acf9ff",
      "family_id": "qf-mod23-multivar-x-slice-table",
      "seed": 202610237073,
      "semantic_template": "qf-mod23-multivar-x-slice-table/{\"coefficients\":[10,0,-2,1,0,-1],\"xs\":[-2,-1,0],\"y\":0}"
    },
    "fields": {
      "title": "A Fixed-Second-Coordinate Slice on Listed Inputs Only",
      "stem": "Use the declared real polynomial R²→R with coefficients=[10, 0, -2, 1, 0, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. fixed y=0, x input list=[-1, 0, -2] normalized to sorted distinct integers. Give (coefficients,fixed y,normalized inputs,every [x,f(x,y)] row). Empty input lists are allowed.",
      "answer": {
        "kind": "expression",
        "value": "([10, 0, -2, 1, 0, -1], 0, [-2, -1, 0], [[-2, 14], [-1, 11], [0, 10]])",
        "explanation": "Fix the second coordinate and evaluate the first-coordinate slice only at listed inputs. A finite or empty slice does not determine the whole function. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([10, 0, -2, 1, 0, -1], 0, [-2, -1, 0], [[-2, 14], [-1, 11], [0, 10]])."
      },
      "skills": [
        "A Fixed-Second-Coordinate Slice on Listed Inputs Only"
      ],
      "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": "明示R2到R多項式與座標角色。操作=multivar-x-slice-table；資料={\"coefficients\":[10,0,-2,1,0,-1],\"y\":0,\"xs\":[-1,0,-2]}。請依 [coefficients,fixed_y,xs,table] 欄序回答。",
        "en": "Use the declared real polynomial R²→R with coefficients=[10, 0, -2, 1, 0, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. fixed y=0, x input list=[-1, 0, -2] normalized to sorted distinct integers. Give (coefficients,fixed y,normalized inputs,every [x,f(x,y)] row). Empty input lists are allowed."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "original six coefficients=[10, 0, -2, 1, 0, -1]",
      "after": "declared scalar R2->R polynomial;finite audits keep coordinate/value roles",
      "relation": "implies",
      "rule_id": "bivariate_explicit_slice_scope",
      "conditions": [],
      "explanation": {
        "zh-Hant": "依明示公式精確算，不由有限點吻合宣稱全域函數恆等、連續性或可微性的新增形式證明。",
        "en": "Evaluate the declared formula exactly.A finite point match is not a new general formal proof of global equivalence,continuity or differentiability."
      },
      "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": "fixed second coordinate y=0;raw first-coordinate list=[-1, 0, -2]",
      "after": "sorted distinct first coordinates=[-2, -1, 0]",
      "relation": "implies",
      "rule_id": "slice_fixed_second_coordinate_not_global_curve",
      "conditions": [],
      "explanation": {
        "zh-Hant": "固定y，只去重排序題目給的x清單；空清單不補點，有限切片不決定整個函數。",
        "en": "Fix y and normalize only the supplied x list.Do not invent points for an empty list or claim a finite slice determines the whole function."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "slice x-2 x squared := mul(-2, -2)",
      "after": "slice x-2 x squared = 4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步原始角色與操作數精確計算，不靠浮點近似或重排原資料。",
        "en": "Compute exactly with this step's original roles and operands,without floating approximation or reordering source data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 slice x-2 x squared，不以小數近似替代。",
          "en": "Recompute slice x-2 x squared with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-2",
          "-2"
        ],
        "output": "4",
        "label": "slice x-2 x squared"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "slice x-2 x times y := mul(-2, 0)",
      "after": "slice x-2 x times y = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步原始角色與操作數精確計算，不靠浮點近似或重排原資料。",
        "en": "Compute exactly with this step's original roles and operands,without floating approximation or reordering source data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 slice x-2 x times y，不以小數近似替代。",
          "en": "Recompute slice x-2 x times y with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-2",
          "0"
        ],
        "output": "0",
        "label": "slice x-2 x times y"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "slice x-2 y squared := mul(0, 0)",
      "after": "slice x-2 y squared = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步原始角色與操作數精確計算，不靠浮點近似或重排原資料。",
        "en": "Compute exactly with this step's original roles and operands,without floating approximation or reordering source data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 slice x-2 y squared，不以小數近似替代。",
          "en": "Recompute slice x-2 y squared with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "0"
        ],
        "output": "0",
        "label": "slice x-2 y squared"
      }
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "slice x-2:x=-2;y=0;coefficients=[10, 0, -2, 1, 0, -1]",
      "after": "fixed monomial values [1,x,y,x^2,xy,y^2]=[1, -2, 0, 4, 0, 0]",
      "relation": "implies",
      "rule_id": "bivariate_fixed_coordinate_monomial_roles",
      "conditions": [],
      "explanation": {
        "zh-Hant": "六個係數固定對應[1,x,y,x²,xy,y²]，不交換x與y，也不當成向量值函數。",
        "en": "The six coefficients correspond to[1,x,y,x²,xy,y²] in that order.Do not exchange x and y or reinterpret the scalar function as vector-valued."
      },
      "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": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "slice x-2 monomial term0 := mul(10, 1)",
      "after": "slice x-2 monomial term0 = 10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步原始角色與操作數精確計算，不靠浮點近似或重排原資料。",
        "en": "Compute exactly with this step's original roles and operands,without floating approximation or reordering source data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 slice x-2 monomial term0，不以小數近似替代。",
          "en": "Recompute slice x-2 monomial term0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "10",
          "1"
        ],
        "output": "10",
        "label": "slice x-2 monomial term0"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "slice x-2 monomial term1 := mul(0, -2)",
      "after": "slice x-2 monomial term1 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步原始角色與操作數精確計算，不靠浮點近似或重排原資料。",
        "en": "Compute exactly with this step's original roles and operands,without floating approximation or reordering source data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 slice x-2 monomial term1，不以小數近似替代。",
          "en": "Recompute slice x-2 monomial term1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "-2"
        ],
        "output": "0",
        "label": "slice x-2 monomial term1"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "slice x-2 monomial term2 := mul(-2, 0)",
      "after": "slice x-2 monomial term2 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步原始角色與操作數精確計算，不靠浮點近似或重排原資料。",
        "en": "Compute exactly with this step's original roles and operands,without floating approximation or reordering source data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 slice x-2 monomial term2，不以小數近似替代。",
          "en": "Recompute slice x-2 monomial term2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-2",
          "0"
        ],
        "output": "0",
        "label": "slice x-2 monomial term2"
      }
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "slice x-2 monomial term3 := mul(1, 4)",
      "after": "slice x-2 monomial term3 = 4",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步原始角色與操作數精確計算，不靠浮點近似或重排原資料。",
        "en": "Compute exactly with this step's original roles and operands,without floating approximation or reordering source data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 slice x-2 monomial term3，不以小數近似替代。",
          "en": "Recompute slice x-2 monomial term3 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "4"
        ],
        "output": "4",
        "label": "slice x-2 monomial term3"
      }
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "slice x-2 monomial term4 := mul(0, 0)",
      "after": "slice x-2 monomial term4 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步原始角色與操作數精確計算，不靠浮點近似或重排原資料。",
        "en": "Compute exactly with this step's original roles and operands,without floating approximation or reordering source data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 slice x-2 monomial term4，不以小數近似替代。",
          "en": "Recompute slice x-2 monomial term4 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "0"
        ],
        "output": "0",
        "label": "slice x-2 monomial term4"
      }
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "slice x-2 monomial term5 := mul(-1, 0)",
      "after": "slice x-2 monomial term5 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步原始角色與操作數精確計算，不靠浮點近似或重排原資料。",
        "en": "Compute exactly with this step's original roles and operands,without floating approximation or reordering source data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 slice x-2 monomial term5，不以小數近似替代。",
          "en": "Recompute slice x-2 monomial term5 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-1",
          "0"
        ],
        "output": "0",
        "label": "slice x-2 monomial term5"
      }
    },
    {
      "id": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "slice x-2 complete polynomial sum := sum(10, 0, 0, 4, 0, 0)",
      "after": "slice x-2 complete polynomial sum = 14",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步原始角色與操作數精確計算，不靠浮點近似或重排原資料。",
        "en": "Compute exactly with this step's original roles and operands,without floating approximation or reordering source data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 slice x-2 complete polynomial sum，不以小數近似替代。",
          "en": "Recompute slice x-2 complete polynomial sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "10",
          "0",
          "0",
          "4",
          "0",
          "0"
        ],
        "output": "14",
        "label": "slice x-2 complete polynomial sum"
      }
    },
    {
      "id": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "slice x-1 x squared := mul(-1, -1)",
      "after": "slice x-1 x squared = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步原始角色與操作數精確計算，不靠浮點近似或重排原資料。",
        "en": "Compute exactly with this step's original roles and operands,without floating approximation or reordering source data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 slice x-1 x squared，不以小數近似替代。",
          "en": "Recompute slice x-1 x squared with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-1",
          "-1"
        ],
        "output": "1",
        "label": "slice x-1 x squared"
      }
    },
    {
      "id": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "slice x-1 x times y := mul(-1, 0)",
      "after": "slice x-1 x times y = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步原始角色與操作數精確計算，不靠浮點近似或重排原資料。",
        "en": "Compute exactly with this step's original roles and operands,without floating approximation or reordering source data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 slice x-1 x times y，不以小數近似替代。",
          "en": "Recompute slice x-1 x times y with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-1",
          "0"
        ],
        "output": "0",
        "label": "slice x-1 x times y"
      }
    },
    {
      "id": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "slice x-1 y squared := mul(0, 0)",
      "after": "slice x-1 y squared = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步原始角色與操作數精確計算，不靠浮點近似或重排原資料。",
        "en": "Compute exactly with this step's original roles and operands,without floating approximation or reordering source data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 slice x-1 y squared，不以小數近似替代。",
          "en": "Recompute slice x-1 y squared with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "0"
        ],
        "output": "0",
        "label": "slice x-1 y squared"
      }
    },
    {
      "id": "S017",
      "uses": [
        "G001",
        "S016"
      ],
      "before": "slice x-1:x=-1;y=0;coefficients=[10, 0, -2, 1, 0, -1]",
      "after": "fixed monomial values [1,x,y,x^2,xy,y^2]=[1, -1, 0, 1, 0, 0]",
      "relation": "implies",
      "rule_id": "bivariate_fixed_coordinate_monomial_roles",
      "conditions": [],
      "explanation": {
        "zh-Hant": "六個係數固定對應[1,x,y,x²,xy,y²]，不交換x與y，也不當成向量值函數。",
        "en": "The six coefficients correspond to[1,x,y,x²,xy,y²] in that order.Do not exchange x and y or reinterpret the scalar function as vector-valued."
      },
      "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": "slice x-1 monomial term0 := mul(10, 1)",
      "after": "slice x-1 monomial term0 = 10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步原始角色與操作數精確計算，不靠浮點近似或重排原資料。",
        "en": "Compute exactly with this step's original roles and operands,without floating approximation or reordering source data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 slice x-1 monomial term0，不以小數近似替代。",
          "en": "Recompute slice x-1 monomial term0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "10",
          "1"
        ],
        "output": "10",
        "label": "slice x-1 monomial term0"
      }
    },
    {
      "id": "S019",
      "uses": [
        "G001",
        "S018"
      ],
      "before": "slice x-1 monomial term1 := mul(0, -1)",
      "after": "slice x-1 monomial term1 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步原始角色與操作數精確計算，不靠浮點近似或重排原資料。",
        "en": "Compute exactly with this step's original roles and operands,without floating approximation or reordering source data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 slice x-1 monomial term1，不以小數近似替代。",
          "en": "Recompute slice x-1 monomial term1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "-1"
        ],
        "output": "0",
        "label": "slice x-1 monomial term1"
      }
    },
    {
      "id": "S020",
      "uses": [
        "G001",
        "S019"
      ],
      "before": "slice x-1 monomial term2 := mul(-2, 0)",
      "after": "slice x-1 monomial term2 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步原始角色與操作數精確計算，不靠浮點近似或重排原資料。",
        "en": "Compute exactly with this step's original roles and operands,without floating approximation or reordering source data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 slice x-1 monomial term2，不以小數近似替代。",
          "en": "Recompute slice x-1 monomial term2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-2",
          "0"
        ],
        "output": "0",
        "label": "slice x-1 monomial term2"
      }
    },
    {
      "id": "S021",
      "uses": [
        "G001",
        "S020"
      ],
      "before": "slice x-1 monomial term3 := mul(1, 1)",
      "after": "slice x-1 monomial term3 = 1",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步原始角色與操作數精確計算，不靠浮點近似或重排原資料。",
        "en": "Compute exactly with this step's original roles and operands,without floating approximation or reordering source data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 slice x-1 monomial term3，不以小數近似替代。",
          "en": "Recompute slice x-1 monomial term3 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "1"
        ],
        "output": "1",
        "label": "slice x-1 monomial term3"
      }
    },
    {
      "id": "S022",
      "uses": [
        "G001",
        "S021"
      ],
      "before": "slice x-1 monomial term4 := mul(0, 0)",
      "after": "slice x-1 monomial term4 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步原始角色與操作數精確計算，不靠浮點近似或重排原資料。",
        "en": "Compute exactly with this step's original roles and operands,without floating approximation or reordering source data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 slice x-1 monomial term4，不以小數近似替代。",
          "en": "Recompute slice x-1 monomial term4 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "0"
        ],
        "output": "0",
        "label": "slice x-1 monomial term4"
      }
    },
    {
      "id": "S023",
      "uses": [
        "G001",
        "S022"
      ],
      "before": "slice x-1 monomial term5 := mul(-1, 0)",
      "after": "slice x-1 monomial term5 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步原始角色與操作數精確計算，不靠浮點近似或重排原資料。",
        "en": "Compute exactly with this step's original roles and operands,without floating approximation or reordering source data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 slice x-1 monomial term5，不以小數近似替代。",
          "en": "Recompute slice x-1 monomial term5 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-1",
          "0"
        ],
        "output": "0",
        "label": "slice x-1 monomial term5"
      }
    },
    {
      "id": "S024",
      "uses": [
        "G001",
        "S023"
      ],
      "before": "slice x-1 complete polynomial sum := sum(10, 0, 0, 1, 0, 0)",
      "after": "slice x-1 complete polynomial sum = 11",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步原始角色與操作數精確計算，不靠浮點近似或重排原資料。",
        "en": "Compute exactly with this step's original roles and operands,without floating approximation or reordering source data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 slice x-1 complete polynomial sum，不以小數近似替代。",
          "en": "Recompute slice x-1 complete polynomial sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "10",
          "0",
          "0",
          "1",
          "0",
          "0"
        ],
        "output": "11",
        "label": "slice x-1 complete polynomial sum"
      }
    },
    {
      "id": "S025",
      "uses": [
        "G001",
        "S024"
      ],
      "before": "slice x0 x squared := mul(0, 0)",
      "after": "slice x0 x squared = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步原始角色與操作數精確計算，不靠浮點近似或重排原資料。",
        "en": "Compute exactly with this step's original roles and operands,without floating approximation or reordering source data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 slice x0 x squared，不以小數近似替代。",
          "en": "Recompute slice x0 x squared with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "0"
        ],
        "output": "0",
        "label": "slice x0 x squared"
      }
    },
    {
      "id": "S026",
      "uses": [
        "G001",
        "S025"
      ],
      "before": "slice x0 x times y := mul(0, 0)",
      "after": "slice x0 x times y = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步原始角色與操作數精確計算，不靠浮點近似或重排原資料。",
        "en": "Compute exactly with this step's original roles and operands,without floating approximation or reordering source data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 slice x0 x times y，不以小數近似替代。",
          "en": "Recompute slice x0 x times y with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "0"
        ],
        "output": "0",
        "label": "slice x0 x times y"
      }
    },
    {
      "id": "S027",
      "uses": [
        "G001",
        "S026"
      ],
      "before": "slice x0 y squared := mul(0, 0)",
      "after": "slice x0 y squared = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步原始角色與操作數精確計算，不靠浮點近似或重排原資料。",
        "en": "Compute exactly with this step's original roles and operands,without floating approximation or reordering source data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 slice x0 y squared，不以小數近似替代。",
          "en": "Recompute slice x0 y squared with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "0"
        ],
        "output": "0",
        "label": "slice x0 y squared"
      }
    },
    {
      "id": "S028",
      "uses": [
        "G001",
        "S027"
      ],
      "before": "slice x0:x=0;y=0;coefficients=[10, 0, -2, 1, 0, -1]",
      "after": "fixed monomial values [1,x,y,x^2,xy,y^2]=[1, 0, 0, 0, 0, 0]",
      "relation": "implies",
      "rule_id": "bivariate_fixed_coordinate_monomial_roles",
      "conditions": [],
      "explanation": {
        "zh-Hant": "六個係數固定對應[1,x,y,x²,xy,y²]，不交換x與y，也不當成向量值函數。",
        "en": "The six coefficients correspond to[1,x,y,x²,xy,y²] in that order.Do not exchange x and y or reinterpret the scalar function as vector-valued."
      },
      "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": "S029",
      "uses": [
        "G001",
        "S028"
      ],
      "before": "slice x0 monomial term0 := mul(10, 1)",
      "after": "slice x0 monomial term0 = 10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步原始角色與操作數精確計算，不靠浮點近似或重排原資料。",
        "en": "Compute exactly with this step's original roles and operands,without floating approximation or reordering source data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 slice x0 monomial term0，不以小數近似替代。",
          "en": "Recompute slice x0 monomial term0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "10",
          "1"
        ],
        "output": "10",
        "label": "slice x0 monomial term0"
      }
    },
    {
      "id": "S030",
      "uses": [
        "G001",
        "S029"
      ],
      "before": "slice x0 monomial term1 := mul(0, 0)",
      "after": "slice x0 monomial term1 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步原始角色與操作數精確計算，不靠浮點近似或重排原資料。",
        "en": "Compute exactly with this step's original roles and operands,without floating approximation or reordering source data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 slice x0 monomial term1，不以小數近似替代。",
          "en": "Recompute slice x0 monomial term1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "0"
        ],
        "output": "0",
        "label": "slice x0 monomial term1"
      }
    },
    {
      "id": "S031",
      "uses": [
        "G001",
        "S030"
      ],
      "before": "slice x0 monomial term2 := mul(-2, 0)",
      "after": "slice x0 monomial term2 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步原始角色與操作數精確計算，不靠浮點近似或重排原資料。",
        "en": "Compute exactly with this step's original roles and operands,without floating approximation or reordering source data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 slice x0 monomial term2，不以小數近似替代。",
          "en": "Recompute slice x0 monomial term2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-2",
          "0"
        ],
        "output": "0",
        "label": "slice x0 monomial term2"
      }
    },
    {
      "id": "S032",
      "uses": [
        "G001",
        "S031"
      ],
      "before": "slice x0 monomial term3 := mul(1, 0)",
      "after": "slice x0 monomial term3 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步原始角色與操作數精確計算，不靠浮點近似或重排原資料。",
        "en": "Compute exactly with this step's original roles and operands,without floating approximation or reordering source data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 slice x0 monomial term3，不以小數近似替代。",
          "en": "Recompute slice x0 monomial term3 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "0"
        ],
        "output": "0",
        "label": "slice x0 monomial term3"
      }
    },
    {
      "id": "S033",
      "uses": [
        "G001",
        "S032"
      ],
      "before": "slice x0 monomial term4 := mul(0, 0)",
      "after": "slice x0 monomial term4 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步原始角色與操作數精確計算，不靠浮點近似或重排原資料。",
        "en": "Compute exactly with this step's original roles and operands,without floating approximation or reordering source data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 slice x0 monomial term4，不以小數近似替代。",
          "en": "Recompute slice x0 monomial term4 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "0"
        ],
        "output": "0",
        "label": "slice x0 monomial term4"
      }
    },
    {
      "id": "S034",
      "uses": [
        "G001",
        "S033"
      ],
      "before": "slice x0 monomial term5 := mul(-1, 0)",
      "after": "slice x0 monomial term5 = 0",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步原始角色與操作數精確計算，不靠浮點近似或重排原資料。",
        "en": "Compute exactly with this step's original roles and operands,without floating approximation or reordering source data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 slice x0 monomial term5，不以小數近似替代。",
          "en": "Recompute slice x0 monomial term5 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-1",
          "0"
        ],
        "output": "0",
        "label": "slice x0 monomial term5"
      }
    },
    {
      "id": "S035",
      "uses": [
        "G001",
        "S034"
      ],
      "before": "slice x0 complete polynomial sum := sum(10, 0, 0, 0, 0, 0)",
      "after": "slice x0 complete polynomial sum = 10",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本步原始角色與操作數精確計算，不靠浮點近似或重排原資料。",
        "en": "Compute exactly with this step's original roles and operands,without floating approximation or reordering source data."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 slice x0 complete polynomial sum，不以小數近似替代。",
          "en": "Recompute slice x0 complete polynomial sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "10",
          "0",
          "0",
          "0",
          "0",
          "0"
        ],
        "output": "10",
        "label": "slice x0 complete polynomial sum"
      }
    }
  ],
  "math_model": {
    "result": [
      [
        10,
        0,
        -2,
        1,
        0,
        -1
      ],
      0,
      [
        -2,
        -1,
        0
      ],
      [
        [
          -2,
          14
        ],
        [
          -1,
          11
        ],
        [
          0,
          10
        ]
      ]
    ]
  },
  "conclusion": {
    "canonical_value": "([10, 0, -2, 1, 0, -1], 0, [-2, -1, 0], [[-2, 14], [-1, 11], [0, 10]])",
    "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-20261013-advance-20261023-r07-multivariable-real-function-foundations",
      "batch_path": "public/data/questions/batches/batch-20261013-advance-20261023-r07-multivariable-real-function-foundations/questions.jsonl",
      "batch_sha256": "bbcd1bbf9a79f292228317e6265f305ff43d9aef65cddda0b0ef95dd02563ab4",
      "original_line": 73,
      "original_line_sha256": "f0cbb1a3e98df7c3042c427d3e4d0b86fd915757c592f6e1dfd3b8b7a35c3300",
      "question_compact_sha256": "e510807f4ff5bb52a28633465a530c0baa4d2df1d3ee17cc434be323f9acf9ff",
      "english_row_compact_sha256": "3b4fd212fe85cd4665be063217c4a8ead3baf5ba8f656d6f01f6bcb0a4f135d6",
      "english_manifest_path": "public/data/i18n/editions/batch-20261013-advance-20261023-r07-multivariable-real-function-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261013-advance-20261023-r07-multivariable-real-function-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-multivariable-real-function": {
        "sha256": "f96ebe8babf93fa8de6b32cb4156ad2b710b938550218a6aff16be79ba4ef928",
        "english_sha256": "3bc9e3338d7c9026e1405d2719cc08c58786f3e4c4480e9803ca45cf10591239",
        "original_path": "public/data/mko/multivariable-real-function.json",
        "english_path": "public/data/i18n/editions/mko-multivariable-real-function.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "bd67681fffd9e29972c322d497cd945a668a794dd25e0c8cd26f1a6ce22e4dfb",
    "original_candidate_sha256": "32710f7e6ecebdb06fbb4344ad192cf12b7279038de9aabbefb06a2385099611",
    "original_candidate_line_sha256": "32710f7e6ecebdb06fbb4344ad192cf12b7279038de9aabbefb06a2385099611"
  },
  "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-20261026-006-p21",
    "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": "b4c47938fd38085d44967226cfa8337d61334455472e7307c60d3a488cef598f",
      "question_first_witness_sha256": "3c23baa97a5933bc07b44f2257017ef4e845bf1429bd6f5381410681c43b97e4"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 35,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "7d5aef5bf4806d0f9c7104a42ba48c10becb1d45579cd62a734b75362cd0ec63",
      "arithmetic_steps": 30
    },
    "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."
  }
}