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```python
# AI Dev Tool: FastAPI Pydantic v2 Micro-SaaS with PostgreSQL Vector Search Extension
# Description: A production-ready starter boilerplate for building high-performance micro-SaaS applications
#              with FastAPI, Pydantic v2, and PostgreSQL Vector Search Extension.

from fastapi import FastAPI, Depends, HTTPException
from fastapi.security import HTTPBearer, HTTPAuthorizationCredentials
from pydantic import BaseModel
from typing import List
import psycopg2
from psycopg2 import Error
import json

# Initialize FastAPI application
app = FastAPI()

# Define security scheme for authentication
security = HTTPBearer()

# Define Pydantic model for user authentication
class UserAuth(BaseModel):
    username: str
    password: str

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

# Define PostgreSQL connection settings
POSTGRES_HOST = "localhost"
POSTGRES_DB = "mydatabase"
POSTGRES_USER = "myuser"
POSTGRES_PASSWORD = "mypassword"

# Establish PostgreSQL connection
def get_postgres_connection():
    try:
        return psycopg2.connect(
            host=POSTGRES_HOST,
            database=

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