{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-finite-group-stabilizer",
  "semantic_family_id": "sf-finite-group-stabilizer",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "有限群作用的穩定子群"
  },
  "summary": {
    "zh-Hant": "指定點的stabilizer是有限群作用中固定該點的全部群元素，並形成子群。"
  },
  "statement": {
    "zh-Hant": "在同一完整認證作用G作用X上，Stab_G(x)={g:g·x=x}須遍歷全部g。完整驗identity、product closure與inverse closure；單一fixed element、action kernel與orbit均不可混同stabilizer。"
  },
  "formula": {
    "tex": "S(x)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>S</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "S",
        "glyph": "S"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "x",
          "glyph": "x"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "4867d055bf11c51859f081f5c2244d2fa0e9d787a53afbd18fd299e2df3f2efe"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "完整有限群表與action table已驗全部作用公理。"
    },
    {
      "id": "a2",
      "zh-Hant": "指定point屬同一action carrier，inverse由同一群表求得。"
    }
  ],
  "symbols": [
    {
      "token": "S",
      "role": "point_stabilizer",
      "role_zh": "點穩定子群",
      "scope": "representative_formula"
    },
    {
      "token": "x",
      "role": "acted_point",
      "role_zh": "指定點",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-finite-group-action",
      "title_zh": "有限群作用",
      "reason_zh": "完整作用模型。"
    },
    {
      "id": "mko-subgroup-homomorphism",
      "title_zh": "子群與群同態",
      "reason_zh": "子群closure語義。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "membership由g·x=x判定。",
      "identity、product與inverse closure皆須完整核。",
      "stabilizer依point改變且不等於orbit或kernel。"
    ],
    "common_misconception_zh": "只找到一個fixed group element就稱已得到stabilizer。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "只處理完整有限作用表。",
    "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 theorem。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-finite-group-stabilizer.md 整理；2026-10-14 脈衝 R01（使用者手動接續 2026-11-09 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"
  }
}