{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-finite-unital-ring",
  "semantic_family_id": "sf-finite-unital-ring",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "有限含一環"
  },
  "summary": {
    "zh-Hant": "有限含一環在同一完整有限載體上具加法阿貝爾群、乘法結合律、左右分配律及與零不同的乘法單位元。"
  },
  "statement": {
    "zh-Hant": "在同一有限載體X上給完整加法表與乘法表。加法須為阿貝爾群；乘法須結合；乘法對加法左右分配；並存在唯一乘法單位元1且1不等於加法單位元0。本輪不要求乘法交換。"
  },
  "formula": {
    "tex": "R(X)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>R</mi><mo>(</mo><mi>X</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "R",
        "glyph": "R"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "X",
          "glyph": "X"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "9fdaabf8d78dd22f9a05b71da0b8373d2f1e689a7142095714667e842068f0bf"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "載體標籤為0到n-1，兩個n乘n運算表皆完整。"
    },
    {
      "id": "a2",
      "zh-Hant": "資格由全部元素對或三元組精確核對；有限抽樣不替代環公理。"
    }
  ],
  "symbols": [
    {
      "token": "R",
      "role": "finite_unital_ring",
      "role_zh": "有限含一環紀錄",
      "scope": "representative_formula"
    },
    {
      "token": "X",
      "role": "finite_carrier",
      "role_zh": "共同有限載體",
      "scope": "representative_formula"
    },
    {
      "token": "0",
      "role": "additive_identity",
      "role_zh": "加法單位元",
      "scope": "representative_formula"
    },
    {
      "token": "1",
      "role": "multiplicative_identity",
      "role_zh": "乘法單位元",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-finite-abelian-group",
      "title_zh": "有限阿貝爾群",
      "reason_zh": "環的加法結構須在同一加法表上成為阿貝爾群。"
    },
    {
      "id": "mko-arithmetic-operations",
      "title_zh": "算術運算",
      "reason_zh": "加法與乘法是共同載體上的二元運算。"
    },
    {
      "id": "mko-function-mapping",
      "title_zh": "映射",
      "reason_zh": "兩個完整表表示X乘X到X的總函數。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "先由加法表找0與加法逆元，再核乘法結合與左右分配。",
      "乘法單位元必須雙側作用且與0不同。",
      "交換環、整環、域、理想與商環另立節點，不由本資格自動推出。"
    ],
    "common_misconception_zh": "把只有乘法單位元的有限表、只通過幾個分配樣本的結構，或乘法交換性當作含一環的全部定義。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "只分類所給完整有限雙表；不外推任意環，也不宣稱交換、無零因子或每個非零元可逆。",
    "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": "加入有限交換環、雙邊理想與商環。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-finite-unital-ring.md 整理；2026-10-14 脈衝 R01（使用者手動接續 2026-10-27 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"
  }
}