{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-real-bilinear-form",
  "semantic_family_id": "sf-real-bilinear-form",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "實雙線性形式"
  },
  "summary": {
    "zh-Hant": "實向量空間上的雙線性形式分別對兩個輸入保持向量加法與實純量乘法。"
  },
  "statement": {
    "zh-Hant": "B:V乘V到R為雙線性，若固定任一輸入後，另一輸入上的映射皆為線性。本輪用整係數矩陣公式B(x,y)=x^T A y作全稱座標證書並精確求值。"
  },
  "formula": {
    "tex": "B(x,y)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>B</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "B",
        "glyph": "B"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "x",
          "glyph": "x"
        },
        {
          "type": "symbol",
          "name": "y",
          "glyph": "y"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "33c43b6959d0d65fa9024b45bb9d6e0cc3bc882849e63864881d860a63ebed54"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "向量與矩陣維度完整相容，座標皆為安全整數。"
    },
    {
      "id": "a2",
      "zh-Hant": "矩陣展開是雙線性的全稱代數證書；有限樣本本身不足以證任意黑箱函數雙線性。"
    }
  ],
  "symbols": [
    {
      "token": "B",
      "role": "real_bilinear_form",
      "role_zh": "實雙線性形式",
      "scope": "representative_formula"
    },
    {
      "token": "A",
      "role": "matrix",
      "role_zh": "座標矩陣",
      "scope": "representative_formula"
    },
    {
      "token": "x",
      "role": "vector",
      "role_zh": "第一輸入",
      "scope": "representative_formula"
    },
    {
      "token": "y",
      "role": "vector",
      "role_zh": "第二輸入",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-real-vector-space",
      "title_zh": "實向量空間",
      "reason_zh": "兩個輸入及線性組合位於實向量空間。"
    },
    {
      "id": "mko-linear-combination-span",
      "title_zh": "有限線性組合與生成",
      "reason_zh": "雙線性分別保持有限線性組合。"
    },
    {
      "id": "mko-function-mapping",
      "title_zh": "映射",
      "reason_zh": "B是V乘V到R的總函數。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "第一槽與第二槽線性責任分開核對。",
      "任意實矩陣都給雙線性形式。",
      "對稱、正定、非退化與內積資格是額外條件。"
    ],
    "common_misconception_zh": "只核B(x,y)=B(y,x)，或幾個樣本的加法，就宣稱雙線性或內積。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "矩陣公式證雙線性但不自動證對稱、正定、非退化或內積。",
    "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": "加入實內積空間及內積誘導範數。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-real-bilinear-form.md 整理；2026-10-14 脈衝 R04（使用者手動接續 2026-10-27 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"
  }
}