{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-finite-group-center",
  "semantic_family_id": "sf-finite-group-center",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "有限群中心"
  },
  "summary": {
    "zh-Hant": "有限群G的中心Z(G)由與每個群元素交換的全部元素構成，且是G的正規子群。"
  },
  "statement": {
    "zh-Hant": "由完整Cayley table建立每個候選元素對所有g的commutation row，Z(G)是所有row全真者。需直接驗identity、product、inverse closure形成subgroup，再對所有g,z驗gzg^{-1}仍在Z(G)以證normal。z的conjugacy class為singleton恰當z∈Z(G)；Z(G)=G恰當G abelian。中心不等於單一元素的centralizer，本輪不宣稱characteristic或class equation。"
  },
  "formula": {
    "tex": "Z(G)=C",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mrow><mi>Z</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><mi>C</mi></mrow></math>",
    "semantic_ast": {
      "type": "equation",
      "lhs": {
        "type": "function_call",
        "callee": {
          "type": "symbol",
          "name": "Z",
          "glyph": "Z"
        },
        "arguments": [
          {
            "type": "symbol",
            "name": "G",
            "glyph": "G"
          }
        ]
      },
      "rhs": {
        "type": "symbol",
        "name": "C",
        "glyph": "C"
      }
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "405967e7d8a00208c9632f84717f9e5917d7c18196d588d6550f750ac2dd2cb4"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "完整有限Cayley table、identity與所有inverse先認證。"
    },
    {
      "id": "a2",
      "zh-Hant": "commutation、subgroup與normality均完整枚舉。"
    }
  ],
  "symbols": [
    {
      "token": "Z",
      "role": "group_center_operator",
      "role_zh": "群中心算子",
      "scope": "representative_formula"
    },
    {
      "token": "G",
      "role": "finite_group",
      "role_zh": "有限群",
      "scope": "representative_formula"
    },
    {
      "token": "C",
      "role": "central_subgroup",
      "role_zh": "中心子群",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-finite-group",
      "title_zh": "有限群",
      "reason_zh": "完整群運算。"
    },
    {
      "id": "mko-finite-group-conjugacy-class",
      "title_zh": "有限群共軛類",
      "reason_zh": "singleton conjugacy class。"
    },
    {
      "id": "mko-finite-normal-subgroup",
      "title_zh": "有限正規子群",
      "reason_zh": "center normality。"
    },
    {
      "id": "mko-quantifier",
      "title_zh": "量詞",
      "reason_zh": "forall g的交換責任。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "完整commutation table定義中心。",
      "subgroup與normality分別稽核。",
      "singleton class、abelian與noncentral witness分開處理。"
    ],
    "common_misconception_zh": "把Z(G)與C_G(x)混同，或只抽樣幾個g便判central。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "不提前宣稱center characteristic或使用class equation。",
    "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class equation與p-group center結果。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-finite-group-center.md 整理；2026-10-15 脈衝 R02（使用者手動接續 2026-11-15 工作序列；實際日期仍為 2026-10-15），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-15"
  }
}