{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261013-advance-20261023-r07-multivariable-real-function-0003",
  "pilot_number": "R0020-10603",
  "number": 10603,
  "family_id": "qf-mod23-multivar-evaluate",
  "title": {
    "zh-Hant": "第 10603 題：二變數實函數",
    "en": "Question 10603: Evaluate with Fixed Bivariate Coefficient Roles"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261013-advance-20261023-r07-multivariable-real-function-0003",
    "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-evaluate；資料={\"coefficients\":[-10,0,-1,1,1,-1],\"x\":0,\"y\":-1}。請依 [coefficients,x,y,value] 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "([-10, 0, -1, 1, 1, -1], 0, -1, -10)",
      "explanation_zh": "現代橋梁核對=([-10, 0, -1, 1, 1, -1], 0, -1, -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": 1,
      "skills": [
        "二變數實函數"
      ]
    },
    "variant": {
      "family_id": "qf-mod23-multivar-evaluate",
      "seed": 202610237003,
      "parameters": {
        "coefficients": [
          -10,
          0,
          -1,
          1,
          1,
          -1
        ],
        "x": 0,
        "y": -1,
        "domain": "Polynomial_R2_to_R_slice",
        "operation": "multivar-evaluate",
        "semantic_template": "qf-mod23-multivar-evaluate/{\"coefficients\":[-10,0,-1,1,1,-1],\"x\":0,\"y\":-1}"
      }
    },
    "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-0003",
    "batch_id": "batch-20261013-advance-20261023-r07-multivariable-real-function-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "32957462f218a6eb3b07458f0f69c24720cc778583e6ac30e91159c7a56e26f8",
      "family_id": "qf-mod23-multivar-evaluate",
      "seed": 202610237003,
      "semantic_template": "qf-mod23-multivar-evaluate/{\"coefficients\":[-10,0,-1,1,1,-1],\"x\":0,\"y\":-1}"
    },
    "fields": {
      "title": "Evaluate with Fixed Bivariate Coefficient Roles",
      "stem": "Use the declared real polynomial R²→R with coefficients=[-10, 0, -1, 1, 1, -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. x=0, y=-1. Give (coefficients,x,y,exact polynomial value).",
      "answer": {
        "kind": "expression",
        "value": "([-10, 0, -1, 1, 1, -1], 0, -1, -10)",
        "explanation": "Substitute each coordinate into its declared monomial; swapping x and y changes the semantic task. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-10, 0, -1, 1, 1, -1], 0, -1, -10)."
      },
      "skills": [
        "Evaluate with Fixed Bivariate Coefficient Roles"
      ],
      "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-evaluate；資料={\"coefficients\":[-10,0,-1,1,1,-1],\"x\":0,\"y\":-1}。請依 [coefficients,x,y,value] 欄序回答。",
        "en": "Use the declared real polynomial R²→R with coefficients=[-10, 0, -1, 1, 1, -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. x=0, y=-1. Give (coefficients,x,y,exact polynomial value)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "original six coefficients=[-10, 0, -1, 1, 1, -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": "query point x squared := mul(0, 0)",
      "after": "query point 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 query point x squared，不以小數近似替代。",
          "en": "Recompute query point x squared with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "0"
        ],
        "output": "0",
        "label": "query point x squared"
      }
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "query point x times y := mul(0, -1)",
      "after": "query point 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 query point x times y，不以小數近似替代。",
          "en": "Recompute query point x times y with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "-1"
        ],
        "output": "0",
        "label": "query point x times y"
      }
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "query point y squared := mul(-1, -1)",
      "after": "query point y 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 query point y squared，不以小數近似替代。",
          "en": "Recompute query point y squared with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-1",
          "-1"
        ],
        "output": "1",
        "label": "query point y squared"
      }
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "query point:x=0;y=-1;coefficients=[-10, 0, -1, 1, 1, -1]",
      "after": "fixed monomial values [1,x,y,x^2,xy,y^2]=[1, 0, -1, 0, 0, 1]",
      "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": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "query point monomial term0 := mul(-10, 1)",
      "after": "query point 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 query point monomial term0，不以小數近似替代。",
          "en": "Recompute query point monomial term0 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-10",
          "1"
        ],
        "output": "-10",
        "label": "query point monomial term0"
      }
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "query point monomial term1 := mul(0, 0)",
      "after": "query point 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 query point monomial term1，不以小數近似替代。",
          "en": "Recompute query point monomial term1 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "0",
          "0"
        ],
        "output": "0",
        "label": "query point monomial term1"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "query point monomial term2 := mul(-1, -1)",
      "after": "query point monomial term2 = 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 query point monomial term2，不以小數近似替代。",
          "en": "Recompute query point monomial term2 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-1",
          "-1"
        ],
        "output": "1",
        "label": "query point monomial term2"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "query point monomial term3 := mul(1, 0)",
      "after": "query point 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 query point monomial term3，不以小數近似替代。",
          "en": "Recompute query point monomial term3 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "0"
        ],
        "output": "0",
        "label": "query point monomial term3"
      }
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "query point monomial term4 := mul(1, 0)",
      "after": "query point 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 query point monomial term4，不以小數近似替代。",
          "en": "Recompute query point monomial term4 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "1",
          "0"
        ],
        "output": "0",
        "label": "query point monomial term4"
      }
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "query point monomial term5 := mul(-1, 1)",
      "after": "query point monomial term5 = -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 query point monomial term5，不以小數近似替代。",
          "en": "Recompute query point monomial term5 with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "mul",
        "inputs": [
          "-1",
          "1"
        ],
        "output": "-1",
        "label": "query point monomial term5"
      }
    },
    {
      "id": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "query point complete polynomial sum := sum(-10, 0, 1, 0, 0, -1)",
      "after": "query point 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 query point complete polynomial sum，不以小數近似替代。",
          "en": "Recompute query point complete polynomial sum with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "-10",
          "0",
          "1",
          "0",
          "0",
          "-1"
        ],
        "output": "-10",
        "label": "query point complete polynomial sum"
      }
    }
  ],
  "math_model": {
    "result": [
      [
        -10,
        0,
        -1,
        1,
        1,
        -1
      ],
      0,
      -1,
      -10
    ]
  },
  "conclusion": {
    "canonical_value": "([-10, 0, -1, 1, 1, -1], 0, -1, -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": 3,
      "original_line_sha256": "e98dd63bae1f78beda975914b4752a5d57364dc4d2accfe1f09e1bcde3dcf6c3",
      "question_compact_sha256": "32957462f218a6eb3b07458f0f69c24720cc778583e6ac30e91159c7a56e26f8",
      "english_row_compact_sha256": "6c046974ecd7ff5eaea760a400c74fb481c5da02bd8521b6c83458c09ce47969",
      "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": "9d1ecd82ebdb9e9f7bc2c261f3880c36c6298607fe78c1dbc88fcf1b31c40456",
    "original_candidate_line_sha256": "9d1ecd82ebdb9e9f7bc2c261f3880c36c6298607fe78c1dbc88fcf1b31c40456"
  },
  "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": 12,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "fcfc1dc1bf5b84b3eab3d05abe9f65dac9467b324cf29c642801a0173daad5f7",
      "arithmetic_steps": 10
    },
    "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."
  }
}