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```python
# AI Dev Tool: FastAPI PostgreSQL Vector Search Extension Boilerplate
# Description: A production-ready boilerplate for building scalable vector search applications
# using FastAPI, PostgreSQL, and Pydantic v2.

from fastapi import FastAPI, Depends
from fastapi.responses import JSONResponse
from pydantic import BaseModel
from typing import List, Optional
from psycopg2 import connect
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 database connection parameters
POSTGRES_HOST = "localhost"
POSTGRES_PORT = 5432
POSTGRES_USER = "postgres"
POSTGRES_PASSWORD = "postgres"
POSTGRES_DB = "vector_search"

# Create a PostgreSQL database engine
engine = create_engine(f"postgresql://{POSTGRES_USER}:{POSTGRES_PASSWORD}@{POSTGRES_HOST}:{POSTGRES_PORT}/{POSTGRES_DB}")

# Create a configured "Session" class
SessionLocal = sessionmaker(autocommit=False, autoflush=False, bind=engine)

# Create a base class for declarative class definitions
Base = declarative_base()

# Define the VectorSearch model

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