{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-vector",
  "semantic_family_id": "sf-vector",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "固定基底下的平面向量"
  },
  "summary": {
    "zh-Hant": "本輪把平面向量限制為固定基底下的 R² 有序對，逐分量定義加法與純量乘法。"
  },
  "statement": {
    "zh-Hant": "在固定標準基底下，平面向量 u=(u₁,u₂)∈R²；u+v=(u₁+v₁,u₂+v₂)，k u=(k u₁,k u₂)，u−v=(u₁−v₁,u₂−v₂)。向量相等當且僅當兩個對應分量相等；零向量為 (0,0)。"
  },
  "formula": {
    "tex": "u+v",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>u</mi><mo>+</mo><mi>v</mi></mrow></math>",
    "semantic_ast": {
      "type": "addition",
      "operands": [
        {
          "type": "symbol",
          "name": "u",
          "glyph": "u"
        },
        {
          "type": "symbol",
          "name": "v",
          "glyph": "v"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "b8b9cb707259354fb16d32ea286074deb8d1cba08429d5cc9e1313f07569b7a0"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "基底固定，分量與純量為實數；題庫採整數分量／純量的精確切片。"
    },
    {
      "id": "a2",
      "zh-Hant": "有序對在此表示向量分量，不把幾何點、位置向量和任意座標系混為同一個對象。"
    }
  ],
  "symbols": [
    {
      "token": "u",
      "role": "plane_vector",
      "role_zh": "第一平面向量",
      "scope": "representative_formula"
    },
    {
      "token": "v",
      "role": "plane_vector",
      "role_zh": "第二平面向量",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-real-number",
      "title_zh": "實數",
      "reason_zh": "向量分量與純量屬於實數。"
    },
    {
      "id": "mko-arithmetic-operations",
      "title_zh": "四則運算",
      "reason_zh": "向量運算按分量做加減與乘法。"
    },
    {
      "id": "mko-coordinate-plane",
      "title_zh": "座標平面",
      "reason_zh": "固定有序分量與基底語境。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "逐分量相加或相減，不能只比較分量和。u−v 的方向有意義；交換通常得到相反向量。",
      "純量零給出零向量，負純量給出方向反轉的代數表示。向量加法與純量乘法不是內積，結果仍是 R² 的向量。",
      "本輪不由分量運算推論長度、角度、單位向量或基底變換；任意向量空間與其公理、物理單位另有義務。"
    ],
    "common_misconception_zh": "把向量座標當成可交換、向量相等只需長度相同，或把純量乘法當成內積。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "整數分量的有限運算可精確重算；一般向量空間公理及幾何語義橋仍未附 Evidence。",
    "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": "綁定 Fin 2→Real／Real×Real 的逐分量結構與固定基底表示。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-vector.md 整理；2026-10-03 脈衝 R05（依使用者要求提前完成 2026-10-04 的流程），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-03"
  }
}