{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-linear-map-kernel",
  "semantic_family_id": "sf-linear-map-kernel",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "線性映射的核"
  },
  "summary": {
    "zh-Hant": "線性映射T的核是全部滿足T(x)=0的來源向量，並形成來源子空間。"
  },
  "statement": {
    "zh-Hant": "本輪以精確2×2整數矩陣綁定T:R²→R²，使用符號null equations或精確row reduction求kernel。bounded coordinate search只回答該盒內，不可取代全實數kernel。非零fiber是affine solution set，不是kernel。"
  },
  "formula": {
    "tex": "K(T)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>K</mi><mo>(</mo><mi>T</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "K",
        "glyph": "K"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "T",
          "glyph": "T"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "2538326014436f33286480b74438f1581d1b33f7c6bd6e0fd2aed14bd91d6704"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "完整2×2矩陣與通常R²線性映射。"
    },
    {
      "id": "a2",
      "zh-Hant": "kernel target固定為零向量；所有除法使用精確有理數。"
    }
  ],
  "symbols": [
    {
      "token": "K",
      "role": "linear_map_kernel",
      "role_zh": "線性映射核",
      "scope": "representative_formula"
    },
    {
      "token": "T",
      "role": "linear_map",
      "role_zh": "指定線性映射",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-general-linear-map",
      "title_zh": "線性映射",
      "reason_zh": "固定T及其線性責任。"
    },
    {
      "id": "mko-vector-subspace",
      "title_zh": "向量子空間",
      "reason_zh": "kernel形成來源子空間。"
    },
    {
      "id": "mko-equation",
      "title_zh": "方程",
      "reason_zh": "解Ax=0。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "membership精確核Ax=0。",
      "完整kernel以null equations或basis表達。",
      "bounded search無解不證全R²無解；非零fiber不是kernel。"
    ],
    "common_misconception_zh": "把Ax=b的非零fiber叫kernel，或由有限網格推全域kernel。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "只處理精確2×2矩陣；bounded結果明示範圍。",
    "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nullity及rank-nullity。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-linear-map-kernel.md 整理；2026-10-14 脈衝 R02（使用者手動接續 2026-11-02 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"
  }
}