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

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```python
# AI Dev Tool: FastAPI Pydantic v2 Micro-SaaS Boilerplate with PostgreSQL Vector Search Extension
# =================================================================================================

from fastapi import FastAPI, Depends, HTTPException
from fastapi.security import OAuth2PasswordBearer, OAuth2PasswordRequestForm
from pydantic import BaseModel, EmailStr
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 IntegrityError
from typing import List, Optional
from psycopg2.extras import Json

# Initialize FastAPI app
app = FastAPI()

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

# PostgreSQL database connection
SQLALCHEMY_DATABASE_URL = "postgresql://user:password@localhost/dbname"
engine = create_engine(SQLALCHEMY_DATABASE_URL)
SessionLocal = sessionmaker(autocommit=False, autoflush=False, bind=engine)

# Define database models
Base = declarative_base()

class User(Base):
    """User model"""
    __tablename__ = "users"
    id = Column(Integer, primary_key=True, index=True)
    username = Column(String, unique=True, index=True)
    email

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