{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-matrix",
  "semantic_family_id": "sf-matrix",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "有限矩陣的尺寸、索引與轉置"
  },
  "summary": {
    "zh-Hant": "矩陣以行、列組織元素；本輪使用非空有限實矩陣及精確整數陣列切片。"
  },
  "statement": {
    "zh-Hant": "m×n 實矩陣 M 是以行 i 與列 j 指定的實數表，m、n 為正整數。本輪外部索引從1開始，M(i,j) 是第i行第j列元素。同尺寸矩陣逐項相加；轉置 Mᵀ 的尺寸是 n×m，並把 M(i,j) 放到 Mᵀ(j,i)。"
  },
  "formula": {
    "tex": "M(i,j)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>M</mi><mo>(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "M",
        "glyph": "M"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "i",
          "glyph": "i"
        },
        {
          "type": "symbol",
          "name": "j",
          "glyph": "j"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "e3470c0824dbc17b01ac213e3423f15d24befca9b8e74bb03d1eeeb89b8a38e3"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "外部行列索引從1開始，不混用程式的零起算陣列索引。"
    },
    {
      "id": "a2",
      "zh-Hant": "題庫每邊尺寸1至5，行長一致且非空，元素為安全整數；零矩陣合法，不處理零行或零列維度。"
    },
    {
      "id": "a3",
      "zh-Hant": "加法先驗尺寸相同；轉置不是逐行倒序，也不是求逆矩陣。"
    }
  ],
  "symbols": [
    {
      "token": "M",
      "role": "real_matrix",
      "role_zh": "指定有限實矩陣",
      "scope": "representative_formula"
    },
    {
      "token": "i",
      "role": "row_index",
      "role_zh": "從1開始的行索引",
      "scope": "representative_formula"
    },
    {
      "token": "j",
      "role": "column_index",
      "role_zh": "從1開始的列索引",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-arithmetic-operations",
      "title_zh": "四則運算",
      "reason_zh": "同位置元素的精確加法。"
    },
    {
      "id": "mko-real-number",
      "title_zh": "實數",
      "reason_zh": "明示元素所在數域。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "矩陣尺寸、某一元素與整個矩陣是不同輸出；同樣的元素重新排位置可形成不同矩陣。尺寸題只以實際行列數區分變種，不靠改無關元素湊題量。",
      "取元素時兩個索引均須落在合法範圍。元素題的身份只保留尺寸、索引及該格資料，不以其他格的改動產生偽變種。",
      "逐項加法只限同尺寸；轉置交換行列，單行會變單列，方陣才保持尺寸。矩陣乘法不是逐項乘法，本輪不把兩者混同。"
    ],
    "common_misconception_zh": "行列顛倒、零起算和一起算混用，或把轉置當成反序或反矩陣。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "有限整數陣列可完整重算；一般矩陣表示及型別索引橋未新增形式 Evidence。",
    "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": "以固定版 Matrix、Fin 索引及 transpose 橋接外部1起算規約，保留正維度切片。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-matrix.md 整理；2026-10-06 脈衝 R01（使用者手動接續 2026-10-07 工作序列；實際日期仍為 2026-10-06），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-06"
  }
}