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

from fastapi import FastAPI, HTTPException
from pydantic import BaseModel
from typing import List, Optional
import psycopg2
import psycopg2.extras
from psycopg2 import Error

# Initialize the FastAPI application
app = FastAPI()

# Define the PostgreSQL database connection settings
POSTGRES_HOST = "localhost"
POSTGRES_DB = "vector_search"
POSTGRES_USER = "postgres"
POSTGRES_PASSWORD = "password"

# Define the Pydantic model for vector search queries
class VectorSearchQuery(BaseModel):
    """Pydantic model for vector search queries"""
    query: str
    limit: int = 10

# Define the Pydantic model for vector search results
class VectorSearchResult(BaseModel):
    """Pydantic model for vector search results"""
    id: int
    vector: List[float]
    score: float

# Establish a connection to the PostgreSQL database
def connect_to_db():
    """Establish a connection to the PostgreSQL

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