{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-gradient-vector",
  "semantic_family_id": "sf-gradient-vector",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "梯度向量"
  },
  "summary": {
    "zh-Hant": "在固定standard Euclidean coordinates中，可微實值函數的一階偏導依座標排列成梯度向量。"
  },
  "statement": {
    "zh-Hant": "對f:R^n→R，gradient為(partial_1 f,...,partial_n f)，座標順序屬資料。本輪exact slice使用有理係數多項式terms [coefficient, exponent-vector]，以symbolic differentiation產生各coordinate polynomial並在有理點精確求值。gradient zero只證stationary，不充分推出local/global extremum；partial derivatives逐一存在在一般情況不保證differentiability。邊界包含norm在原點的missing partial、x^2-y^2的saddle、不完整coordinate list，以及f(x,y)=s*x^3/(x^2+y^2)、f(0,0)=0在原點兩偏導存在但沿y=x否證可微。"
  },
  "formula": {
    "tex": "G(f,x)=v",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mrow><mi>G</mi><mo>(</mo><mi>f</mi><mo>,</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mi>v</mi></mrow></math>",
    "semantic_ast": {
      "type": "equation",
      "lhs": {
        "type": "function_call",
        "callee": {
          "type": "symbol",
          "name": "G",
          "glyph": "G"
        },
        "arguments": [
          {
            "type": "symbol",
            "name": "f",
            "glyph": "f"
          },
          {
            "type": "symbol",
            "name": "x",
            "glyph": "x"
          }
        ]
      },
      "rhs": {
        "type": "symbol",
        "name": "v",
        "glyph": "v"
      }
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "1058bfe2211ed39b7e0855f1dba847750e59e1ac24a81fa3147cca48ae6236f7"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "dimension、term exponents與point arity完全一致；同類monomials先canonical合併。"
    },
    {
      "id": "a2",
      "zh-Hant": "座標基底固定為standard Euclidean basis，不靜默變換metric或coordinates。"
    }
  ],
  "symbols": [
    {
      "token": "G",
      "role": "gradient_operator",
      "role_zh": "梯度算子",
      "scope": "representative_formula"
    },
    {
      "token": "f",
      "role": "multivariable_real_function",
      "role_zh": "多變數實函數",
      "scope": "representative_formula"
    },
    {
      "token": "x",
      "role": "coordinate_point",
      "role_zh": "座標點",
      "scope": "representative_formula"
    },
    {
      "token": "v",
      "role": "coordinate_gradient_vector",
      "role_zh": "梯度向量",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-partial-derivative",
      "title_zh": "偏導數",
      "reason_zh": "coordinate derivatives。"
    },
    {
      "id": "mko-vector",
      "title_zh": "向量",
      "reason_zh": "ordered component vector。"
    },
    {
      "id": "mko-multivariable-real-function",
      "title_zh": "多變數實函數",
      "reason_zh": "domain與scalar codomain。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "symbolic gradient保留dimension與coordinate order。",
      "point evaluation使用exact rationals。",
      "stationary、differentiable與extremum是不同責任。"
    ],
    "common_misconception_zh": "把partial存在當可微，或把gradient zero當極小／global optimum證明。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "本輪不自動給gradient descent收斂、Lagrange multiplier或Hessian分類。",
    "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": "加入梯度下降與Lagrange乘數必要條件。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-gradient-vector.md 整理；2026-10-15 脈衝 R06（使用者手動接續 2026-11-13 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"
  }
}