{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-maximum-matching",
  "semantic_family_id": "sf-maximum-matching",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "最大匹配"
  },
  "summary": {
    "zh-Hant": "最大匹配是在完整有限圖全部合法matching中邊數最大者；可有多個且不必perfect。"
  },
  "statement": {
    "zh-Hant": "本輪對小型完整有限簡單圖exhaustively枚舉全部edge subsets並篩matching，給exact maximum size與witnesses。每個候選先驗matching；maximal不等於maximum，多個maximum可共存。"
  },
  "formula": {
    "tex": "X(G)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>X</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "X",
        "glyph": "X"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "G",
          "glyph": "G"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "cdbc27594c65cc0bd4e9e898eaecf78aae159d3841b227b5f99e3e592a267b5f"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "完整有限無向無loop簡單圖，edge count受合約上限。"
    },
    {
      "id": "a2",
      "zh-Hant": "全域結論依完整2^|E| enumeration或等價global upper bound。"
    }
  ],
  "symbols": [
    {
      "token": "X",
      "role": "maximum_matching",
      "role_zh": "最大匹配",
      "scope": "representative_formula"
    },
    {
      "token": "G",
      "role": "simple_graph",
      "role_zh": "有限簡單圖",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-graph-matching",
      "title_zh": "圖匹配",
      "reason_zh": "合法matching定義。"
    },
    {
      "id": "mko-optimization-problem",
      "title_zh": "最佳化問題",
      "reason_zh": "全域最大基數。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "先核matching合法。",
      "maximum由全域比較／upper bound。",
      "maximal、perfect、unique皆分開。"
    ],
    "common_misconception_zh": "把不能再加邊的maximal當maximum，或把maximum自動當perfect/unique。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "只處理edge count有界的完整有限圖。",
    "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Hall定理與matching algorithms。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-maximum-matching.md 整理；2026-10-14 脈衝 R06（使用者手動接續 2026-11-03 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"
  }
}