{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-real-vector-space",
  "semantic_family_id": "sf-real-vector-space",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "實向量空間"
  },
  "summary": {
    "zh-Hant": "實向量空間配備向量加法與實數純量乘法，滿足加法交換群及相容的純量作用公理。"
  },
  "statement": {
    "zh-Hant": "實向量空間V具有向量加法、零向量、加法逆元與實數純量乘法，並滿足結合、交換、分配、單位純量及純量乘法相容等公理。本輪題庫只核固定維度實座標空間的精確公理實例。"
  },
  "formula": {
    "tex": "V(R)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>V</mi><mo>(</mo><mi>R</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "V",
        "glyph": "V"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "R",
          "glyph": "R"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "c18cfb2835bfda8fee58174917b71551ac2bb1d066cb96721e575fb9d184bc3f"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "題庫使用相同固定維度的整數座標向量與整數純量，視為實數特例。"
    },
    {
      "id": "a2",
      "zh-Hant": "座標實例通過不等於任意候選結構已完整證成向量空間。"
    }
  ],
  "symbols": [
    {
      "token": "V",
      "role": "real_vector_space",
      "role_zh": "實向量空間",
      "scope": "representative_formula"
    },
    {
      "token": "R",
      "role": "scalar_field",
      "role_zh": "實數純量域",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-vector",
      "title_zh": "向量",
      "reason_zh": "座標向量提供具體模型。"
    },
    {
      "id": "mko-real-number",
      "title_zh": "實數",
      "reason_zh": "純量來自實數。"
    },
    {
      "id": "mko-arithmetic-operations",
      "title_zh": "四則運算",
      "reason_zh": "座標加法與純量乘法逐分量計算。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "向量加法只在相同維度間定義，逐分量相加。",
      "純量乘法逐分量縮放；零向量與加法逆元保證加法群結構。",
      "分配律可在指定向量與純量上精確重算，但一般公理仍需對全部元素量化。"
    ],
    "common_misconception_zh": "把不同維度向量直接相加，或用少數例子冒充一般公理證明。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "只核固定R^n座標切片，不宣稱任意抽象結構已證為向量空間。",
    "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": "加入線性組合、生成、獨立、基底與維度。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-real-vector-space.md 整理；2026-10-13 脈衝 R04（使用者手動接續 2026-10-22 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"
  }
}