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```python
# AI Dev Tool: PostgreSQL Vector Search Extension with FastAPI and Supabase
# ========================================================================
# This is a high-demand production-ready developer code asset for building
# a scalable and efficient vector search system using PostgreSQL and FastAPI.

from fastapi import FastAPI, HTTPException
from fastapi.responses import JSONResponse
from fastapi.requests import Request
from supabase import create_client, Client
from typing import List, Dict
import psycopg2
import json

# Initialize FastAPI app
app = FastAPI()

# Initialize Supabase client
SUPABASE_URL = "https://your-supabase-instance.supabase.co"
SUPABASE_KEY = "your-supabase-key"
SUPABASE_SECRET = "your-supabase-secret"
supabase: Client = create_client(SUPABASE_URL, SUPABASE_KEY)

# PostgreSQL connection settings
POSTGRES_HOST = "your-postgres-host"
POSTGRES_PORT = 5432
POSTGRES_USER = "your-postgres-user"
POSTGRES_PASSWORD = "your-postgres-password"
POSTGRES_DB = "your-postgres-db"

# Create a PostgreSQL connection pool
def get_postgres_connection():
    """Returns a PostgreSQL connection object"""
    try:
        conn = psycopg2.connect(

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