{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-vector-subspace",
  "semantic_family_id": "sf-vector-subspace",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "實向量子空間"
  },
  "summary": {
    "zh-Hant": "實向量空間的非空子集若含零向量並對向量加法與每個實純量乘法封閉，則為向量子空間。"
  },
  "statement": {
    "zh-Hant": "本輪只處理R²中的零空間、全空間、過原點直線與明示有限反例。正例使用齊次線性方程或span的全稱代數證書；有限抽樣通過不足以證子空間。"
  },
  "formula": {
    "tex": "U(V)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>U</mi><mo>(</mo><mi>V</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "U",
        "glyph": "U"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "V",
          "glyph": "V"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "a678b59985cfff4ab1c459589d69d23e37dd0bf11948c0fe104188aec070ca6e"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "母空間固定為R²並使用通常向量運算。"
    },
    {
      "id": "a2",
      "zh-Hant": "正例證書須全稱涵蓋加法及任意實純量；仿射直線不經原點不是子空間。"
    }
  ],
  "symbols": [
    {
      "token": "U",
      "role": "vector_subspace",
      "role_zh": "候選子空間",
      "scope": "representative_formula"
    },
    {
      "token": "V",
      "role": "real_vector_space",
      "role_zh": "母向量空間",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-real-vector-space",
      "title_zh": "實向量空間",
      "reason_zh": "提供R²向量運算與純量。"
    },
    {
      "id": "mko-subset",
      "title_zh": "子集合",
      "reason_zh": "子空間是母空間子集。"
    },
    {
      "id": "mko-quantifier",
      "title_zh": "量詞",
      "reason_zh": "封閉性是全稱責任。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "含零向量是必要條件。",
      "過原點齊次直線可由線性方程給全稱封閉證書。",
      "一個反例足以否證；有限正樣本不足以證明。"
    ],
    "common_misconception_zh": "把任意直線或有限抽樣封閉當成子空間證書。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "只在R²與明示證書範圍重算，不外推任意維或未知運算。",
    "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": "加入核、像、維度與商向量空間。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-vector-subspace.md 整理；2026-10-14 脈衝 R01（使用者手動接續 2026-11-01 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"
  }
}