⚡ Live Sandbox: AI Dev Tool: PostgreSQL Vector Search Extension

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

from fastapi import FastAPI, Depends, HTTPException
from fastapi.security import OAuth2PasswordBearer, OAuth2PasswordRequestForm
from pydantic import BaseModel, Field
from pydantic.errors import ListError
from typing import List, Optional
from sqlalchemy import create_engine, Column, Integer, String
from sqlalchemy.ext.declarative import declarative_base
from sqlalchemy.orm import sessionmaker, Session
from sqlalchemy.exc import SQLAlchemyError
from psycopg2.extras import Json
import json
import os

# Initialize the FastAPI application
app = FastAPI()

# Define the OAuth2 password bearer
oauth2_scheme = OAuth2PasswordBearer(tokenUrl="token")

# Define the PostgreSQL database connection
SQLALCHEMY_DATABASE_URL = os.environ.get("DATABASE_URL")
engine = create_engine(SQLALCHEMY_DATABASE_URL)
SessionLocal = sessionmaker(autocommit=False, autoflush=False, bind=engine)
Base = declarative_base()

# Define the user model
class User(Base

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