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```python
# AI Dev Tool: FastAPI Pydantic v2 Micro-SaaS Boilerplate with PostgreSQL Vector Search Extension
# =================================================================================================
# This is a high-demand, production-ready developer code asset for building micro-SaaS applications
# using FastAPI, Pydantic v2, and PostgreSQL Vector Search Extension.

from fastapi import FastAPI, Depends
from fastapi.responses import JSONResponse
from fastapi.requests import Request
from pydantic import BaseModel, Field
from typing import List, Optional
from psycopg2 import connect, Error
from psycopg2.extras import DictCursor
from psycopg2.extensions import ISOLATION_LEVEL_SERIALIZABLE
import os
import logging

# Initialize the FastAPI application
app = FastAPI()

# Define the PostgreSQL database connection settings
POSTGRES_HOST = os.environ.get("POSTGRES_HOST", "localhost")
POSTGRES_PORT = os.environ.get("POSTGRES_PORT", 5432)
POSTGRES_USER = os.environ.get("POSTGRES_USER", "postgres")
POSTGRES_PASSWORD = os.environ.get("POSTGRES_PASSWORD", "postgres")
POSTGRES_DB = os.environ.get("POSTGRES_DB", "postgres")

# Define the PostgreSQL Vector Search Extension table
VECTOR_SEARCH_TABLE = "vector_search"

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