{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-cauchy-theorem-finite-groups",
  "semantic_family_id": "sf-cauchy-theorem-finite-groups",
  "type": "theorem",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "有限群的Cauchy定理"
  },
  "summary": {
    "zh-Hant": "若質數p整除有限群G的階，則G含有一個exact order為p的元素。"
  },
  "statement": {
    "zh-Hant": "G須由完整Cayley table認證為有限群，p須為質數且p整除|G|。結論是存在g使g^p=e且對每個1≤k<p皆有g^k≠e，故ord(g)=p。本輪完整有限枚舉只為指定model產生element-order witness，不取代一般定理證明；不得外推為每個合數d整除|G|就必有order-d元素。"
  },
  "formula": {
    "tex": "o(g)=p",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mrow><mi>o</mi><mo>(</mo><mi>g</mi><mo>)</mo></mrow><mo>=</mo><mi>p</mi></mrow></math>",
    "semantic_ast": {
      "type": "equation",
      "lhs": {
        "type": "function_call",
        "callee": {
          "type": "symbol",
          "name": "o",
          "glyph": "o"
        },
        "arguments": [
          {
            "type": "symbol",
            "name": "g",
            "glyph": "g"
          }
        ]
      },
      "rhs": {
        "type": "symbol",
        "name": "p",
        "glyph": "p"
      }
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "761518c66a6a5d408a33b5bce733993624b662fee1b26294e5a2b66d03d5bd2b"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "正向題族使用完整、封閉且通過結合律、單位元與逆元核對的有限Cayley table。"
    },
    {
      "id": "a2",
      "zh-Hant": "元素階以從1起的最小正整數冪回到identity定義，並完整核對所有較小正冪。"
    }
  ],
  "symbols": [
    {
      "token": "o",
      "role": "element_order",
      "role_zh": "元素階",
      "scope": "representative_formula"
    },
    {
      "token": "g",
      "role": "finite_group_element",
      "role_zh": "有限群元素",
      "scope": "representative_formula"
    },
    {
      "token": "p",
      "role": "prime_divisor",
      "role_zh": "群階的質因數",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-lagrange-theorem-finite-groups",
      "title_zh": "有限群的Lagrange定理",
      "reason_zh": "元素生成的循環子群階整除群階。"
    },
    {
      "id": "mko-prime-number",
      "title_zh": "自然數的質數",
      "reason_zh": "p的質數資格。"
    },
    {
      "id": "mko-finite-group",
      "title_zh": "有限群",
      "reason_zh": "完整群運算。"
    },
    {
      "id": "mko-finite-group-action",
      "title_zh": "有限群作用",
      "reason_zh": "一般證明可使用群作用與軌道計數。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "定理的量詞是每個prime divisor都至少有一個order-p witness。",
      "order-p證書同時要求p次冪回到identity及所有較小正冪不回到identity。",
      "finite table exhaustion只證給定群；一般Cauchy定理仍是定理層責任。"
    ],
    "common_misconception_zh": "只找到g^p=e卻未排除較小階，或把prime divisor結論錯推到任意composite divisor。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "本輪不以有限模型取代一般證明，也不聲稱每個合數因子都是某元素的階。",
    "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Sylow定理與有限群結構推論。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-cauchy-theorem-finite-groups.md 整理；2026-10-14 脈衝 R01（使用者手動接續 2026-11-12 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"
  }
}