{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-finite-group-action",
  "semantic_family_id": "sf-finite-group-action",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "有限群作用"
  },
  "summary": {
    "zh-Hant": "有限群在有限集合上的作用是滿足單位與乘法相容律的total action table。"
  },
  "statement": {
    "zh-Hant": "以同一完整有限群Cayley table、完整point carrier及total action table，全稱核e·x=x與(gh)·x=g·(h·x)。每個g因此誘導排列；有效作用本身不推出faithful、free或transitive，有限抽樣pairs不足。"
  },
  "formula": {
    "tex": "A(g,x)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>A</mi><mo>(</mo><mi>g</mi><mo>,</mo><mi>x</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "A",
        "glyph": "A"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "g",
          "glyph": "g"
        },
        {
          "type": "symbol",
          "name": "x",
          "glyph": "x"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "da4b658117f30560729a16cf9b9e2c159df5a3327ec679c68dba29c400e33dbf"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "群表完整並精確驗closure、associativity、identity與inverse。"
    },
    {
      "id": "a2",
      "zh-Hant": "作用表對每個group element與point恰有一個carrier內值。"
    }
  ],
  "symbols": [
    {
      "token": "A",
      "role": "finite_group_action",
      "role_zh": "有限群作用",
      "scope": "representative_formula"
    },
    {
      "token": "g",
      "role": "group_element",
      "role_zh": "群元素",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-finite-group",
      "title_zh": "有限群",
      "reason_zh": "完整群運算。"
    },
    {
      "id": "mko-function-mapping",
      "title_zh": "函數與映射",
      "reason_zh": "total action map。"
    },
    {
      "id": "mko-set",
      "title_zh": "集合",
      "reason_zh": "被作用carrier。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "單位作用與乘法相容是兩項獨立責任。",
      "有限完整表可逐三元組精確驗證。",
      "faithful、free與transitive需要另立定義與證書。"
    ],
    "common_misconception_zh": "只測幾個(g,x)或只見每列像排列便稱為群作用。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "只處理完整有限表；不推無限群、連續作用或表示理論性質。",
    "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orbit、stabilizer與orbit-stabilizer theorem。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-finite-group-action.md 整理；2026-10-14 脈衝 R01（使用者手動接續 2026-11-07 工作序列；實際日期仍為 2026-10-14），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-14"
  }
}