{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-finite-group-conjugacy-class",
  "semantic_family_id": "sf-finite-group-conjugacy-class",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "有限群共軛類"
  },
  "summary": {
    "zh-Hant": "有限群G以conjugation g·x=gxg^{-1}作用於自身，其orbits形成共軛類並分割G。"
  },
  "statement": {
    "zh-Hant": "必須從完整Cayley table與inverse建立完整conjugation action table，逐一驗identity/action composition。指定x的class是所有gxg^{-1}的完整集合；全部classes兩兩相等或不交且union為G。singleton class恰表示x與所有g commute。明示y不在class(x)即可證兩元素nonconjugate。共軛類不是coset或conjugate subgroup；本輪不使用class-size=[G:C_G(x)]，因此不引入centralizer或額外orbit-stabilizer結論。"
  },
  "formula": {
    "tex": "C(G,x)=O",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mrow><mi>C</mi><mo>(</mo><mi>G</mi><mo>,</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mi>O</mi></mrow></math>",
    "semantic_ast": {
      "type": "equation",
      "lhs": {
        "type": "function_call",
        "callee": {
          "type": "symbol",
          "name": "C",
          "glyph": "C"
        },
        "arguments": [
          {
            "type": "symbol",
            "name": "G",
            "glyph": "G"
          },
          {
            "type": "symbol",
            "name": "x",
            "glyph": "x"
          }
        ]
      },
      "rhs": {
        "type": "symbol",
        "name": "O",
        "glyph": "O"
      }
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "37b1086f3219a3238e5f35f54bcddd6265a2f75cabf59cac5be17a72f03d54fa"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "完整有限Cayley table、identity與所有inverse均先認證。"
    },
    {
      "id": "a2",
      "zh-Hant": "action table、orbits與partition均完整枚舉。"
    }
  ],
  "symbols": [
    {
      "token": "C",
      "role": "conjugacy_class_operator",
      "role_zh": "共軛類算子",
      "scope": "representative_formula"
    },
    {
      "token": "G",
      "role": "finite_group",
      "role_zh": "有限群",
      "scope": "representative_formula"
    },
    {
      "token": "x",
      "role": "group_element",
      "role_zh": "群元素",
      "scope": "representative_formula"
    },
    {
      "token": "O",
      "role": "conjugation_orbit",
      "role_zh": "共軛軌道",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-finite-group-action",
      "title_zh": "有限群作用",
      "reason_zh": "conjugation action公理。"
    },
    {
      "id": "mko-finite-group-orbit",
      "title_zh": "有限群作用的軌道",
      "reason_zh": "orbit partition。"
    },
    {
      "id": "mko-finite-group",
      "title_zh": "有限群",
      "reason_zh": "特殊共軛作用的乘法與inverse。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "conjugation action須由群運算完整導出。",
      "每個class是指定元素的完整orbit。",
      "singleton與nonconjugate witnesses不引入centralizer計數。"
    ],
    "common_misconception_zh": "把共軛類當left coset，或把gHg^{-1}的subgroup conjugate與element class混同。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "不宣稱class-size公式、群中心或類方程；這些留待後續節點。",
    "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": "加入有限群中心與class equation。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-finite-group-conjugacy-class.md 整理；2026-10-15 脈衝 R03（使用者手動接續 2026-11-14 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"
  }
}