{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-multivariable-real-function",
  "semantic_family_id": "sf-multivariable-real-function",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "多變數實函數"
  },
  "summary": {
    "zh-Hant": "多變數實函數把明示實向量域中的輸入映到實數；座標角色與定義域是函數資料。"
  },
  "statement": {
    "zh-Hant": "本條目先限二變數實多項式f:R²→R的明示切片，係數依[1,x,y,x²,xy,y²]固定次序。函數資格需明示定義域與座標角色；本輪不自動宣稱連續、可微或向量值。"
  },
  "formula": {
    "tex": "f(x,y)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>f</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "f",
        "glyph": "f"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "x",
          "glyph": "x"
        },
        {
          "type": "symbol",
          "name": "y",
          "glyph": "y"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "611fd5e7e553a5ef2cb4397365bce1f5cb86e2140460e291853e8327a8c53ab8"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "題庫使用六個整數係數及整數座標點。"
    },
    {
      "id": "a2",
      "zh-Hant": "矩形domain audit只核明示閉矩形；多項式運算雖可全域定義，題目語境仍以宣告domain為準。"
    }
  ],
  "symbols": [
    {
      "token": "f",
      "role": "multivariable_real_function",
      "role_zh": "二變數實函數",
      "scope": "representative_formula"
    },
    {
      "token": "x",
      "role": "first_coordinate",
      "role_zh": "第一座標",
      "scope": "representative_formula"
    },
    {
      "token": "y",
      "role": "second_coordinate",
      "role_zh": "第二座標",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-function-mapping",
      "title_zh": "函數映射",
      "reason_zh": "每個合法輸入對應唯一實值。"
    },
    {
      "id": "mko-vector",
      "title_zh": "向量",
      "reason_zh": "多變數輸入可視為實座標向量。"
    },
    {
      "id": "mko-coordinate-plane",
      "title_zh": "座標平面",
      "reason_zh": "二變數輸入以有序座標表示。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "係數順序固定才能精確重算。",
      "rectangle domain需分開核x與y邊界。",
      "固定y的x-slice只是單一切片，不代表完整函數已由有限表決定。"
    ],
    "common_misconception_zh": "交換x與y係數，或以幾個點值宣稱全域函數等價、連續或可微。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "只核R²到R多項式切片；未建立偏導數或一般正則性。",
    "evidence_refs": [],
    "producers": [
      {
        "id": "lean-mathlib",
        "adapter": "formal_system",
        "config_path": "formal/lean/lakefile.toml",
        "status": "configured"
      }
    ]
  },
  "formalization": {
    "status": "not_formalized",
    "target_systems": [
      "Lean 4",
      "Mathlib"
    ],
    "next_obligation_zh": "加入偏導數、梯度與PDE。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-multivariable-real-function.md 整理；2026-10-13 脈衝 R07（使用者手動接續 2026-10-23 工作序列；實際日期仍為 2026-10-13），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-13"
  }
}