{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-inner-product-induced-norm",
  "semantic_family_id": "sf-inner-product-induced-norm",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "內積誘導範數"
  },
  "summary": {
    "zh-Hant": "同一實內積以向量自內積的非負平方根定義範數；三角不等式依Cauchy–Schwarz全稱證書。"
  },
  "statement": {
    "zh-Hant": "在同一實內積空間中定義norm(x)=sqrt(<x,x>)。正定性給分離，齊次性給絕對齊次，Cauchy–Schwarz推出三角不等式。本輪以R²正定矩陣精確計算平方範數與恆等式，不以樣本取代全稱證書。"
  },
  "formula": {
    "tex": "N(J)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>N</mi><mo>(</mo><mi>J</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "N",
        "glyph": "N"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "J",
          "glyph": "J"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "788d32ec5bbf256b472f0a05a551090af2901b94ba708a05b3c6091411cc6e4f"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "內積矩陣已通過雙線性、對稱與正定，向量位於同一R²。"
    },
    {
      "id": "a2",
      "zh-Hant": "平方根保持精確符號；Cauchy–Schwarz是全稱證書，不由有限向量樣本建立。"
    }
  ],
  "symbols": [
    {
      "token": "N",
      "role": "inner_product_induced_norm",
      "role_zh": "內積誘導範數",
      "scope": "representative_formula"
    },
    {
      "token": "J",
      "role": "real_inner_product",
      "role_zh": "同一實內積",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-real-inner-product-space",
      "title_zh": "實內積空間",
      "reason_zh": "誘導公式使用同一已認證內積。"
    },
    {
      "id": "mko-power-root",
      "title_zh": "冪與根式",
      "reason_zh": "範數為自內積的非負平方根。"
    },
    {
      "id": "mko-order-relation",
      "title_zh": "序關係",
      "reason_zh": "正定性與Cauchy–Schwarz使用實數順序。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "平方範數<x,x>先以精確整數計算。",
      "距離平方由<x-y,x-y>給出。",
      "Cauchy–Schwarz與parallelogram identity提供結構核對，但不推出完備。"
    ],
    "common_misconception_zh": "看到若干向量三角不等式通過就宣稱一般範數，或由內積誘導範數直接叫Hilbert。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "只核明示R²內積矩陣與精確證書；不推出完備或Hilbert。",
    "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": "與完備性共同加入實Hilbert空間。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-inner-product-induced-norm.md 整理；2026-10-14 脈衝 R03（使用者手動接續 2026-10-29 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"
  }
}