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```python
# AI Dev Tool: FastAPI Pydantic v2 Micro-SaaS with PostgreSQL Vector Search Extension
# ==============================================================================
# This is a production-ready developer code asset for building high-performance Micro-SaaS applications.
# It combines the power of FastAPI, Pydantic v2, and PostgreSQL Vector Search Extension.

from fastapi import FastAPI, HTTPException
from pydantic import BaseModel
from typing import List, Optional
import psycopg2
import psycopg2.extras
from psycopg2 import Error

# Initialize the FastAPI application
app = FastAPI(title="FastAPI Pydantic v2 Micro-SaaS",
              description="A high-performance Micro-SaaS application with PostgreSQL Vector Search Extension",
              version="1.0.0")

# Define the PostgreSQL database connection settings
POSTGRES_HOST = "localhost"
POSTGRES_PORT = 5432
POSTGRES_DB = "mydatabase"
POSTGRES_USER = "myuser"
POSTGRES_PASSWORD = "mypassword"

# Define the Pydantic model for the search query
class SearchQuery(BaseModel):
    """The search query model"""
    query: str
    limit: int = 10
    offset: int = 0

# Define the Pydantic model for

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