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```python
# AI Dev Tool: FastAPI Pydantic v2 Micro-SaaS Boilerplate with PostgreSQL Vector Search Extension
# ==================================================================================================
# This is a high-demand, production-ready developer code asset for building micro-SaaS applications
# using FastAPI, Pydantic v2, and PostgreSQL Vector Search Extension.

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

# Initialize the FastAPI application
app = FastAPI()

# Define the PostgreSQL database connection settings
POSTGRES_HOST = "localhost"
POSTGRES_PORT = 5432
POSTGRES_USER = "postgres"
POSTGRES_PASSWORD = "postgres"
POSTGRES_DB = "micro_saaS"

# Define the Pydantic model for the micro-SaaS application
class User(BaseModel):
    id: int
    name: str
    email: str

# Define the Pydantic model for the vector search query
class VectorSearchQuery(BaseModel):
    query: str
    limit: int = 10

# Create a connection to the PostgreSQL database
def create_connection():
    try:
        connection

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