{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-linear-map-image",
  "semantic_family_id": "sf-linear-map-image",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "線性映射的像"
  },
  "summary": {
    "zh-Hant": "線性映射的像是所有輸出T(x)的集合；矩陣映射的像等於column span。"
  },
  "statement": {
    "zh-Hant": "本輪用精確2×2矩陣與symbolic column-span certificate。membership以Ax=y的精確一致性判定；bounded fiber只提供witness或bounded nonexistence，不可證全R²非membership。image、codomain與指定map range不可混同。"
  },
  "formula": {
    "tex": "I(T)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>I</mi><mo>(</mo><mi>T</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "I",
        "glyph": "I"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "T",
          "glyph": "T"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "6116557541d195ac885e5eed37aa0fcc62c576f0aaae40b5b112153a732cf80f"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "完整2×2矩陣定義T:R²→R²。"
    },
    {
      "id": "a2",
      "zh-Hant": "全域membership使用符號線性系統；bounded search另標界線。"
    }
  ],
  "symbols": [
    {
      "token": "I",
      "role": "linear_map_image",
      "role_zh": "線性映射像",
      "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。"
    },
    {
      "id": "mko-vector-subspace",
      "title_zh": "向量子空間",
      "reason_zh": "image為目標子空間。"
    },
    {
      "id": "mko-function-mapping",
      "title_zh": "函數映射",
      "reason_zh": "image與codomain區分。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "image為columns的全實數span。",
      "membership等價於Ax=y有實解。",
      "bounded search失敗不是全域nonmembership。"
    ],
    "common_misconception_zh": "把codomain直接當image，或由有限輸入樣本宣稱range完整。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "只處理精確2×2矩陣與明示symbolic certificate。",
    "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及rank-nullity。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-linear-map-image.md 整理；2026-10-14 脈衝 R03（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"
  }
}