{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-direct-sum",
  "semantic_family_id": "sf-direct-sum",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "子空間直和"
  },
  "summary": {
    "zh-Hant": "二個子空間之和若每個向量有唯一分解便是內直和；要宣稱覆蓋母空間還須另證生成。"
  },
  "statement": {
    "zh-Hant": "在有限維實向量空間以完整基底與exact rank核對二項內直和。U交W為零只保證U+W內的唯一性；V=U直和W還需U+W=V。三項以上不得以兩兩交集為零代替整體唯一性，外直和則使用乘積型新載體。"
  },
  "formula": {
    "tex": "U+W",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>U</mi><mo>+</mo><mi>W</mi></mrow></math>",
    "semantic_ast": {
      "type": "addition",
      "operands": [
        {
          "type": "symbol",
          "name": "U",
          "glyph": "U"
        },
        {
          "type": "symbol",
          "name": "W",
          "glyph": "W"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "d9ebc82196d69e560c216ca4b7de0e7ed8d12a6a195048fa5ef2e1e5fbf09a29"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "所有子空間由完整線性獨立基底表示並使用精確有理rank。"
    },
    {
      "id": "a2",
      "zh-Hant": "內直和、覆蓋母空間、多項直和與外直和分開核對。"
    }
  ],
  "symbols": [
    {
      "token": "U",
      "role": "vector_subspace",
      "role_zh": "第一子空間",
      "scope": "representative_formula"
    },
    {
      "token": "W",
      "role": "vector_subspace",
      "role_zh": "第二子空間",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-vector-subspace",
      "title_zh": "實向量子空間",
      "reason_zh": "母空間中的封閉子集。"
    },
    {
      "id": "mko-set-operations",
      "title_zh": "集合運算",
      "reason_zh": "交集與和空間。"
    },
    {
      "id": "mko-linear-independence-basis",
      "title_zh": "線性獨立與基底",
      "reason_zh": "完整基底與rank證書。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "二項內直和的唯一性等價於交集只有零向量。",
      "覆蓋V是另一責任；唯一性本身只談U+W。",
      "三項以上須檢查全部基底聯集的整體獨立性，外直和不可與內直和混同。"
    ],
    "common_misconception_zh": "把U交W為零直接說成V=U直和W，或用三項兩兩交集為零證整體直和。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "只處理有限維實座標與exact rank；不宣稱無限直和或拓撲直和。",
    "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": "加入商空間與短正合列中的分裂。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-direct-sum.md 整理；2026-10-14 脈衝 R01（使用者手動接續 2026-11-05 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"
  }
}