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```python
# AI Dev Tool: PostgreSQL Vector Search Extension with FastAPI and Hono
# ================================================================
# This is a production-ready developer code asset that integrates 
# PostgreSQL Vector Search Extension with FastAPI and Hono Cloudflare Workers.

from fastapi import FastAPI, HTTPException
from fastapi.responses import JSONResponse
from fastapi.requests import Request
from pydantic import BaseModel
from typing import List, Optional
import psycopg2
import json
from hono import Hono

# Define the FastAPI application
app = FastAPI()

# Define the Hono Cloudflare Workers instance
hono = Hono()

# Define the PostgreSQL database connection settings
POSTGRES_HOST = "localhost"
POSTGRES_PORT = 5432
POSTGRES_DB = "vector_search_db"
POSTGRES_USER = "vector_search_user"
POSTGRES_PASSWORD = "vector_search_password"

# Define the PostgreSQL Vector Search Extension query function
def query_vector_search(query: str, limit: int = 10):
    """
    Query the PostgreSQL Vector Search Extension.

Args:
    - query (str): The search query.
    - limit (int): The number of results to return (default: 10).

Returns:
    - A list of search results

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