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```python
# AI Dev Tool: PostgreSQL Vector Search Extension with FastAPI and Supabase
# ==============================================================================
# This is a production-ready developer code asset that integrates PostgreSQL Vector Search Extension
# with FastAPI and Supabase to provide a scalable and efficient search functionality.

import os
import uvicorn
from fastapi import FastAPI, HTTPException
from fastapi.responses import JSONResponse
from fastapi.requests import Request
from supabase import create_client, Client
from typing import Dict, List
from pydantic import BaseModel

# Supabase URL and key
SUPABASE_URL = os.environ.get("SUPABASE_URL")
SUPABASE_KEY = os.environ.get("SUPABASE_KEY")

# Create a Supabase client
supabase: Client = create_client(SUPABASE_URL, SUPABASE_KEY)

# Define the FastAPI app
app = FastAPI()

# Define a Pydantic model for search queries
class SearchQuery(BaseModel):
    query: str
    limit: int = 10

# Define a Pydantic model for search results
class SearchResult(BaseModel):
    id: int
    vector: List[float]
    metadata: Dict[str, str]

# Define the PostgreSQL Vector Search Extension function
async

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