{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-rank-nullity-theorem",
  "semantic_family_id": "sf-rank-nullity-theorem",
  "type": "theorem",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "秩－零度定理"
  },
  "summary": {
    "zh-Hant": "有限維線性映射T:V→W滿足dim V=dim ker T+rank T。"
  },
  "statement": {
    "zh-Hant": "矩陣A為m×n並明示T:R^n→R^m。以exact RREF、kernel basis與column-space rank驗n=nullity+rank。nullity是kernel dimension而非解數；sampled vectors不足。"
  },
  "formula": {
    "tex": "N(T)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>N</mi><mo>(</mo><mi>T</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "N",
        "glyph": "N"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "T",
          "glyph": "T"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "e260dffc79085d815db6e51a1e23cea7a40266e45c6d434d412c5520d0dfef73"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "有限維來源且矩陣shape明示。"
    },
    {
      "id": "a2",
      "zh-Hant": "所有rank/kernel/image證書使用精確有理運算。"
    }
  ],
  "symbols": [
    {
      "token": "N",
      "role": "rank_nullity_record",
      "role_zh": "秩零度紀錄",
      "scope": "representative_formula"
    },
    {
      "token": "T",
      "role": "linear_map",
      "role_zh": "線性映射",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-linear-map-kernel",
      "title_zh": "線性映射核",
      "reason_zh": "nullity來源。"
    },
    {
      "id": "mko-linear-map-image",
      "title_zh": "線性映射像",
      "reason_zh": "rank來源。"
    },
    {
      "id": "mko-matrix-rank",
      "title_zh": "矩陣秩",
      "reason_zh": "精確rank。"
    },
    {
      "id": "mko-vector-space-dimension",
      "title_zh": "向量空間維度",
      "reason_zh": "source/kernel維度。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "source dimension是column count n。",
      "nullity為kernel basis大小。",
      "完整symbolic證書驗identity；sampling不足。"
    ],
    "common_misconception_zh": "把nullity當解的個數，或搞反m×n map方向。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "只處理小型精確矩陣；不由有限sample證一般定理。",
    "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first isomorphism theorem。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-rank-nullity-theorem.md 整理；2026-10-14 脈衝 R01（使用者手動接續 2026-11-04 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"
  }
}