{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-mathematical-representation",
  "semantic_family_id": "sf-mathematical-representation",
  "type": "concept",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "數學表徵"
  },
  "summary": {
    "zh-Hant": "數學表徵把同一對象的文字規則、符號式、有限表格、序對與座標資料分開記錄，並以明示語義橋核對一致性。"
  },
  "statement": {
    "zh-Hant": "同一數學對象可以用文字、符號、表格、序對或圖形表示。兩個表示是否對應，必須先固定對象、定義域、標度與解讀規則；不能只依外觀宣稱等價。"
  },
  "formula": {
    "tex": "R(f)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>R</mi><mo>(</mo><mi>f</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "R",
        "glyph": "R"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "f",
          "glyph": "f"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "2b73f9e3a0ffe690653054191584fd900f43866601b99bcf0d24dafc4ba17508"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "題庫使用整係數仿射規則及完整明示的有限表格／序對。"
    },
    {
      "id": "a2",
      "zh-Hant": "座標或表格資料只在列出的定義域核對；未列點與圖形外觀不供隱含數值。"
    }
  ],
  "symbols": [
    {
      "token": "R",
      "role": "representation_record",
      "role_zh": "表徵紀錄",
      "scope": "representative_formula"
    },
    {
      "token": "f",
      "role": "represented_object",
      "role_zh": "被表示的數學對象",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-variable-expression",
      "title_zh": "變數與代數式",
      "reason_zh": "符號表徵使用含變數代數式。"
    },
    {
      "id": "mko-function-mapping",
      "title_zh": "函數映射",
      "reason_zh": "函數表格需保證每個輸入只有一個輸出。"
    },
    {
      "id": "mko-coordinate-plane",
      "title_zh": "座標平面",
      "reason_zh": "序對與座標表示使用有序分量。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "表格列出指定輸入與輸出，符號式則給計算規則；二者只有在共同域及每列數值相符時形成明示語義橋。",
      "兩點可決定非垂直仿射規則，但必須先確認橫坐標不同；重複輸入若對應不同輸出，就不是函數表。",
      "圖像可以支援理解，但像素位置、視覺直線感或未標示刻度不能替代明示資料。"
    ],
    "common_misconception_zh": "只因圖形看似相同就宣稱公式等價，或忽略定義域與重複輸入衝突。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "有限表格一致性只涵蓋明示域；不推論未列輸入或教學效果。",
    "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": "建立更完整的圖形、語言、符號與資料表之間的 typed semantic bridge。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-mathematical-representation.md 整理；2026-10-13 脈衝 R01（使用者手動接續 2026-10-21 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"
  }
}