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

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```python
# AI Dev Tool: FastAPI Pydantic v2 Micro-SaaS with PostgreSQL Vector Search Extension
# ==============================================================================
# This is a high-demand, production-ready developer code asset for building
# 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, Field
from typing import List, Optional
from sqlalchemy import create_engine, Column, Integer, String
from sqlalchemy.orm import sessionmaker, Session
from sqlalchemy.ext.declarative import declarative_base
from psycopg2.extras import Json

# Initialize FastAPI application
app = FastAPI()

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

# Define Pydantic models
class User(BaseModel):
    id: int = Field(..., description="User ID")
    name: str = Field(..., description="User name")
    email: str = Field

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