{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-real-inner-product-space",
  "semantic_family_id": "sf-real-inner-product-space",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "實內積空間"
  },
  "summary": {
    "zh-Hant": "對稱且正定的實雙線性形式構成實內積；半正定或抽樣為正不足。"
  },
  "statement": {
    "zh-Hant": "在同一實向量空間上，內積須對兩槽雙線性、對稱，且對每個非零向量x都有<x,x>>0。本輪對R^2使用整係數對稱矩陣及Sylvester條件作全稱正定證書。"
  },
  "formula": {
    "tex": "J(V)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>J</mi><mo>(</mo><mi>V</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "J",
        "glyph": "J"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "V",
          "glyph": "V"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "ddd852ca5d686234008e3a16823419ffa3b26782c801b5c1a75a7255f9fecab1"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "矩陣與向量完整明示；候選矩陣為2乘2整數矩陣。"
    },
    {
      "id": "a2",
      "zh-Hant": "正定性以首主子式與行列式的精確條件核對，不由有限向量樣本建立。"
    }
  ],
  "symbols": [
    {
      "token": "J",
      "role": "real_inner_product",
      "role_zh": "實內積",
      "scope": "representative_formula"
    },
    {
      "token": "V",
      "role": "real_vector_space",
      "role_zh": "實向量空間",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-real-bilinear-form",
      "title_zh": "實雙線性形式",
      "reason_zh": "內積先須完整雙線性。"
    },
    {
      "id": "mko-order-relation",
      "title_zh": "序關係",
      "reason_zh": "正定性使用實數嚴格正值。"
    },
    {
      "id": "mko-real-vector-space",
      "title_zh": "實向量空間",
      "reason_zh": "內積定義在同一實向量空間。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "雙線性、對稱、正定是三個分離責任。",
      "R^2對稱矩陣可用Sylvester條件作全稱證書。",
      "內積空間不自動完備。"
    ],
    "common_misconception_zh": "把對稱雙線性形式或半正定形式直接叫內積，或用幾個向量樣本證正定。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "只核明示R^2矩陣證書；不推出完備、Hilbert或任意維譜定理。",
    "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": "加入內積誘導範數。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-real-inner-product-space.md 整理；2026-10-14 脈衝 R04（使用者手動接續 2026-10-28 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"
  }
}