⚡ Live Sandbox: AI Dev Tool: DeepSeek V4 Flash 0731

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```python
# AI Dev Tool: PostgreSQL Vector Search Extension with FastAPI and Supabase
# =====================================================================
# This production-ready developer code asset provides a starter boilerplate
# for building scalable and efficient vector search applications using
# PostgreSQL, FastAPI, and Supabase.

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

# Initialize FastAPI application
app = FastAPI()

# Initialize Supabase client
SUPABASE_URL = os.environ.get("SUPABASE_URL")
SUPABASE_KEY = os.environ.get("SUPABASE_KEY")
supabase: Client = create_client(SUPABASE_URL, SUPABASE_KEY)

# Define PostgreSQL vector search extension
@app.post("/vector_search")
async def vector_search(query: Dict):
    """
    Perform vector search on PostgreSQL database using the provided query.

Args:
        query (Dict): Query dictionary containing the search query and parameters.

Returns:
        JSONResponse: Search results in JSON format.
    """
    try:
        # Extract search query and parameters from the request body
        search_query = query.get("

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