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

from fastapi import FastAPI, Depends
from fastapi.responses import JSONResponse
from fastapi.requests import Request
from pydantic import BaseModel
from typing import List, Optional
from psycopg2 import connect, Error
from psycopg2.extras import DictCursor
from sqlalchemy import create_engine, Column, Integer, String
from sqlalchemy.ext.declarative import declarative_base
from sqlalchemy.orm import sessionmaker, Session

# Define the FastAPI application
app = FastAPI()

# Define the PostgreSQL connection parameters
POSTGRES_HOST = "localhost"
POSTGRES_DB = "vector_search"
POSTGRES_USER = "postgres"
POSTGRES_PASSWORD = "postgres"

# Define the PostgreSQL connection engine
engine = create_engine(f"postgresql://{POSTGRES_USER}:{POSTGRES_PASSWORD}@{POSTGRES_HOST}/{POSTGRES_DB}")

# Define the base class for SQLAlchemy models
Base = declarative_base()

# Define the VectorSearch model
class VectorSearch(Base):
    __tablename__ = "vector_search"
    id = Column(Integer, primary_key=True)

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