{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-finite-abelian-group",
  "semantic_family_id": "sf-finite-abelian-group",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "有限阿貝爾群"
  },
  "summary": {
    "zh-Hant": "有限群若完整Cayley表對所有元素對都滿足交換律，則為有限阿貝爾群。"
  },
  "statement": {
    "zh-Hant": "有限阿貝爾群先須以完整Cayley表通過封閉、結合、唯一單位元與逆元資格，再對所有a,b核a*b=b*a。本輪只分類明示有限表。"
  },
  "formula": {
    "tex": "A(G)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>A</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "A",
        "glyph": "A"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "G",
          "glyph": "G"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "a9cded30cf0885bcd509b5b7b6059b966944bf67128d5d6a7cd59d3b9995d6b2"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "載體標籤為0到n-1且提供完整n乘n表。"
    },
    {
      "id": "a2",
      "zh-Hant": "題族不由幾個樣本或生成元看似交換外推整群交換。"
    }
  ],
  "symbols": [
    {
      "token": "A",
      "role": "abelian_group_record",
      "role_zh": "阿貝爾群紀錄",
      "scope": "representative_formula"
    },
    {
      "token": "G",
      "role": "finite_group",
      "role_zh": "有限群",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-finite-group",
      "title_zh": "有限群",
      "reason_zh": "交換律只在已驗群結構上附加。"
    },
    {
      "id": "mko-quantifier",
      "title_zh": "量詞",
      "reason_zh": "阿貝爾性量化所有元素對。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "完整群資格先於交換性分類。",
      "元素階與倍數由同一群運算及單位元導出。",
      "單一元素生成的循環子群只屬指定有限群。"
    ],
    "common_misconception_zh": "只因幾個元素交換或表對角線對稱就宣稱整群阿貝爾。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "完整有限表只證指定有限群；不外推任意群。",
    "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": "加入有限含一環的加法阿貝爾群與第二運算。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-finite-abelian-group.md 整理；2026-10-13 脈衝 R01（使用者手動接續 2026-10-26 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"
  }
}