{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-vector-space-dimension",
  "semantic_family_id": "sf-vector-space-dimension",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "有限維向量空間的維度"
  },
  "summary": {
    "zh-Hant": "有限維向量空間的維度是任一有限基底的元素數；基底不同但基數不變。"
  },
  "statement": {
    "zh-Hant": "本輪只處理R²/R³完整座標族。先精確核線性獨立與生成，成立後才以基底大小給dimension。獨立族只給下界，生成族只給上界；一般基底基數不變依有限維定理。"
  },
  "formula": {
    "tex": "D(V)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>D</mi><mo>(</mo><mi>V</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "D",
        "glyph": "D"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "V",
          "glyph": "V"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "4ed427b5a21af7be3ab1740e11c4695d11959cfd20d689c537c01134901659f2"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "母空間為R²或R³且向量完整列出。"
    },
    {
      "id": "a2",
      "zh-Hant": "rank以精確有理Gaussian elimination計算；不使用浮點近似。"
    }
  ],
  "symbols": [
    {
      "token": "D",
      "role": "vector_space_dimension",
      "role_zh": "維度紀錄",
      "scope": "representative_formula"
    },
    {
      "token": "V",
      "role": "finite_dimensional_real_vector_space",
      "role_zh": "有限維實向量空間",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-real-vector-space",
      "title_zh": "實向量空間",
      "reason_zh": "提供座標與向量運算。"
    },
    {
      "id": "mko-linear-independence-basis",
      "title_zh": "線性獨立與基底",
      "reason_zh": "基底同時獨立與生成。"
    },
    {
      "id": "mko-cardinality-comparison",
      "title_zh": "集合基數比較",
      "reason_zh": "基底大小與上下界。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "真正基底成立後才能讀取dimension。",
      "獨立族大小不超過dimension。",
      "生成族大小不小於dimension；基底基數不變是定理責任。"
    ],
    "common_misconception_zh": "把任意獨立族或生成族的大小直接當成dimension。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "只重算R²/R³精確矩陣；不以兩個例子證一般basis invariance。",
    "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rank、nullity與rank-nullity theorem。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-vector-space-dimension.md 整理；2026-10-14 脈衝 R01（使用者手動接續 2026-11-02 工作序列；實際日期仍為 2026-10-14），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-14"
  }
}