{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-system-linear-equations",
  "semantic_family_id": "sf-system-linear-equations",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "二元聯立一次方程組與退化邊界"
  },
  "summary": {
    "zh-Hant": "同時滿足兩個實數線性限制才是解；非零行列式給唯一解，奇異與零列須另判一致性。"
  },
  "statement": {
    "zh-Hant": "令 x、y 為實數未知數，要求 a x+b y=c 與 d x+e y=f 同時成立。令 L(x,y)=(a x+b y−c,d x+e y−f)，z=(0,0)，則解集滿足 L(x,y)=z。D=ae−bd≠0 時唯一解為 x=(ce−bf)/D、y=(af−cd)/D；D=0 時須檢查一致性，可能無解或無限多解。"
  },
  "formula": {
    "tex": "L(x,y)=z",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mrow><mi>L</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow><mo>=</mo><mi>z</mi></mrow></math>",
    "semantic_ast": {
      "type": "equation",
      "lhs": {
        "type": "function_call",
        "callee": {
          "type": "symbol",
          "name": "L",
          "glyph": "L"
        },
        "arguments": [
          {
            "type": "symbol",
            "name": "x",
            "glyph": "x"
          },
          {
            "type": "symbol",
            "name": "y",
            "glyph": "y"
          }
        ]
      },
      "rhs": {
        "type": "symbol",
        "name": "z",
        "glyph": "z"
      }
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "3a46e988cbbc91201488da7be94ba8647c293f2f1e049ca4599af58f2121cef7"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "係數與右側常數皆為實數；題庫採整數係數和精確有理候選。"
    },
    {
      "id": "a2",
      "zh-Hant": "作為線性系統容許零係數列：0=0 無限制，0=c≠0 不一致；不得在 D=0 時套唯一解除法。"
    }
  ],
  "symbols": [
    {
      "token": "L",
      "role": "paired_linear_constraint_residual",
      "role_zh": "兩個線性限制的殘差向量",
      "scope": "representative_formula"
    },
    {
      "token": "x",
      "role": "first_real_unknown",
      "role_zh": "第一實數未知數",
      "scope": "representative_formula"
    },
    {
      "token": "y",
      "role": "second_real_unknown",
      "role_zh": "第二實數未知數",
      "scope": "representative_formula"
    },
    {
      "token": "z",
      "role": "zero_vector",
      "role_zh": "零向量 (0,0)",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-linear-equation",
      "title_zh": "一元一次方程式",
      "reason_zh": "非零係數、代入驗證及退化限制需分清。"
    },
    {
      "id": "mko-vector",
      "title_zh": "固定基底下的平面向量",
      "reason_zh": "解與殘差各用有序二分量表示。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "驗證候選有理數對時，必須代入兩列，分母先確認非零；只通過其中一列不等於整個系統的解。",
      "D≠0 時才使用唯一解公式。D=0 時，係數秩與增廣資料的一致性決定無解或無限多解；必須先檢查不一致零列。係數全零且右側全零時所有 R² 皆為解；若任一右側非零則無解。",
      "本輪用精確二階 minors 與零列邊界檢查一致性。不能僅因 D 和兩個 Cramer 分子都為零就認定無限多解，也不從示意直線交點猜精確解。"
    ],
    "common_misconception_zh": "只代入一列、在 D=0 時除法，或忽略 0=非零 的不一致零列。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "明示二元兩列系統可精確驗證；一般線性代數秩定理與求解演算法的形式正確性尚待 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": "綁定線性限制解集、零列、二階 determinant、rank／augmented consistency 與非零 D 的解公式。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-system-linear-equations.md 整理；2026-10-03 脈衝 R06（依使用者要求提前完成 2026-10-04 的流程），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-03"
  }
}