{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-matrix-rank",
  "semantic_family_id": "sf-matrix-rank",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "矩陣秩"
  },
  "summary": {
    "zh-Hant": "矩陣秩是row/column space維度，可由精確rank-preserving消去的pivot數計算。"
  },
  "statement": {
    "zh-Hant": "本輪處理2×2或3×3整數矩陣。非零r-minor只給rank下界；rank上界須全部較大minor為零或完整exact RREF。det=0只排除full rank，不單獨決定rank；row-rank=column-rank依定理。"
  },
  "formula": {
    "tex": "R(A)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>R</mi><mo>(</mo><mi>A</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "R",
        "glyph": "R"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "A",
          "glyph": "A"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "ddebdfb47e19647667f6c2aa52046d41a51465954b3b35fa0bc33605a75ea941"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "矩陣完整列出並使用精確有理Gaussian elimination。"
    },
    {
      "id": "a2",
      "zh-Hant": "所有row operations明示為rank-preserving。"
    }
  ],
  "symbols": [
    {
      "token": "R",
      "role": "matrix_rank",
      "role_zh": "矩陣秩",
      "scope": "representative_formula"
    },
    {
      "token": "A",
      "role": "matrix",
      "role_zh": "指定矩陣",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-matrix",
      "title_zh": "矩陣",
      "reason_zh": "完整有限矩陣。"
    },
    {
      "id": "mko-linear-combination-span",
      "title_zh": "線性組合與span",
      "reason_zh": "row/column spaces。"
    },
    {
      "id": "mko-vector-space-dimension",
      "title_zh": "向量空間維度",
      "reason_zh": "秩為span維度。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "完整RREF的pivot數給rank。",
      "minor證書區分上下界。",
      "det=0不等於rank=0；row/column rank相等是定理責任。"
    ],
    "common_misconception_zh": "只看det=0就宣稱確切rank，或由抽樣row operations給上界。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "只處理2×2/3×3精確矩陣。",
    "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rank-nullity與矩陣分解。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-matrix-rank.md 整理；2026-10-14 脈衝 R01（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"
  }
}