{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-lagrange-multiplier-condition",
  "semantic_family_id": "sf-lagrange-multiplier-condition",
  "type": "theorem",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "等式約束Lagrange乘數必要條件"
  },
  "summary": {
    "zh-Hant": "正則等式約束局部極值點的目標梯度，必為constraint gradients的線性組合。"
  },
  "statement": {
    "zh-Hant": "考慮minimize f(x) subject to g_1(x)=...=g_m(x)=0。若x* feasible、f與g_i在鄰域可微，且rows grad g_i(x*)線性獨立（Jacobian full row rank），則local extremum必要地存在唯一lambda使grad f(x*)=J_g(x*)^T lambda；本條目固定這個正號convention。本輪exact slice用有理向量與rank elimination核regularity、stationarity residual，並完整解單一線性constraint下的squared-distance問題。此方程是必要非充分條件；rank deficient只表示本定理不適用，不表示無極值。它不等於含不等式的KKT，也不推出global optimum或unique primal point。"
  },
  "formula": {
    "tex": "L(f,g,x)=l",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mrow><mi>L</mi><mo>(</mo><mi>f</mi><mo>,</mo><mi>g</mi><mo>,</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mi>l</mi></mrow></math>",
    "semantic_ast": {
      "type": "equation",
      "lhs": {
        "type": "function_call",
        "callee": {
          "type": "symbol",
          "name": "L",
          "glyph": "L"
        },
        "arguments": [
          {
            "type": "symbol",
            "name": "f",
            "glyph": "f"
          },
          {
            "type": "symbol",
            "name": "g",
            "glyph": "g"
          },
          {
            "type": "symbol",
            "name": "x",
            "glyph": "x"
          }
        ]
      },
      "rhs": {
        "type": "symbol",
        "name": "l",
        "glyph": "l"
      }
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "3b1d8beb09f4de70f0e0a92f5386d7136d590c72c907a8e71ad2bd7cef4cb960"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "constraint gradients以rows依宣告順序排列；m不大於dimension且exact rational rank為m。"
    },
    {
      "id": "a2",
      "zh-Hant": "stationarity先要求feasibility與regularity；本輪vector arithmetic使用exact rationals。"
    },
    {
      "id": "a3",
      "zh-Hant": "single-constraint solved slice要求constraint gradient非零，objective是到center的squared distance。"
    }
  ],
  "symbols": [
    {
      "token": "L",
      "role": "lagrange_multiplier_condition",
      "role_zh": "Lagrange乘數條件",
      "scope": "representative_formula"
    },
    {
      "token": "f",
      "role": "objective_function",
      "role_zh": "目標函數",
      "scope": "representative_formula"
    },
    {
      "token": "g",
      "role": "equality_constraint_map",
      "role_zh": "等式約束映射",
      "scope": "representative_formula"
    },
    {
      "token": "x",
      "role": "regular_feasible_point",
      "role_zh": "正則可行點",
      "scope": "representative_formula"
    },
    {
      "token": "l",
      "role": "multiplier_vector",
      "role_zh": "乘數向量",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-gradient-vector",
      "title_zh": "梯度向量",
      "reason_zh": "目標與constraint gradients。"
    },
    {
      "id": "mko-optimization-problem",
      "title_zh": "最佳化問題",
      "reason_zh": "feasible local extremum。"
    },
    {
      "id": "mko-system-linear-equations",
      "title_zh": "二元一次聯立方程組",
      "reason_zh": "multiplier linear system背景。"
    },
    {
      "id": "mko-matrix-rank",
      "title_zh": "矩陣秩",
      "reason_zh": "Jacobian full row rank。"
    },
    {
      "id": "mko-linear-independence-basis",
      "title_zh": "線性獨立與基底",
      "reason_zh": "constraint gradient regularity。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "feasibility、differentiability與full-row-rank資格先行。",
      "固定grad f=J_g^T lambda，避免符號與row/column convention漂移。",
      "necessary stationarity不等於sufficient、KKT、global或unique。"
    ],
    "common_misconception_zh": "解出lambda就宣稱最小值，或在constraint gradients相依時仍聲稱本regular theorem已證。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "本輪不處理inequality KKT、second-order sufficient conditions或global optimality。",
    "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tangent-space proof、KKT constraint qualifications與second-order tests。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-lagrange-multiplier-condition.md 整理；2026-10-15 脈衝 R04（使用者手動接續 2026-11-19 工作序列；實際日期仍為 2026-10-15），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-15"
  }
}