{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-finite-convex-combination",
  "semantic_family_id": "sf-finite-convex-combination",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "有限凸組合"
  },
  "summary": {
    "zh-Hant": "有限凸組合用非負精確權重加權有限非空向量列，且權重總和恰為一。"
  },
  "statement": {
    "zh-Hant": "向量v_1,...,v_n的有限凸組合為sum lambda_i v_i，其中n≥1、每個lambda_i≥0且sum lambda_i=1。零權重允許；負權重或只近似和為1只形成一般仿射或線性組合。"
  },
  "formula": {
    "tex": "C(V)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>C</mi><mo>(</mo><mi>V</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "C",
        "glyph": "C"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "V",
          "glyph": "V"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "d5fc381683e96e8a5be6eb901d77dc5124781953491cd8746a7ec714df44df99"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "題庫用整數座標向量與精確有理權重，不接受浮點近似。"
    },
    {
      "id": "a2",
      "zh-Hant": "凸組合witness只證給定加權點；不主張座標唯一性，除非另有仿射獨立條件。"
    }
  ],
  "symbols": [
    {
      "token": "C",
      "role": "convex_combination_record",
      "role_zh": "凸組合紀錄",
      "scope": "representative_formula"
    },
    {
      "token": "lambda",
      "role": "exact_weight",
      "role_zh": "精確權重",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-real-vector-space",
      "title_zh": "實向量空間",
      "reason_zh": "向量加法與實純量乘法承載組合。"
    },
    {
      "id": "mko-ratio-proportion",
      "title_zh": "比與比例",
      "reason_zh": "有理權重及總和條件。"
    },
    {
      "id": "mko-arithmetic-operations",
      "title_zh": "四則運算",
      "reason_zh": "精確加權與約分。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "資格分別核非負性與總和一。",
      "兩點凸組合以lambda與1-lambda形成線段點。",
      "多點witness以精確重建目標核對。"
    ],
    "common_misconception_zh": "允許負權重、把0.999...近似當精確1，或由凸組合直接聲稱唯一座標。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "只處理有限非空向量列與精確有理權重。",
    "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": "加入凸集合、凸包與仿射獨立。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-finite-convex-combination.md 整理；2026-10-13 脈衝 R05（使用者手動接續 2026-10-24 工作序列；實際日期仍為 2026-10-13），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-13"
  }
}