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```python
# AI Dev Tool: PostgreSQL Vector Search Extension with FastAPI Pydantic v2 Micro-SaaS
# Description: A production-ready developer code asset for building scalable and efficient vector search systems
# Category: Production Starter Boilerplate

from fastapi import FastAPI, HTTPException
from pydantic import BaseModel
from typing import List, Optional
import psycopg2
from psycopg2 import Error
import numpy as np
from scipy import spatial

# Define the FastAPI application
app = FastAPI()

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

# Define the Pydantic model for vector search requests
class VectorSearchRequest(BaseModel):
    """Vector search request model"""
    query_vector: List[float]
    top_k: Optional[int] = 10

# Define the Pydantic model for vector search responses
class VectorSearchResponse(BaseModel):
    """Vector search response model"""
    results: List[dict]

# Establish a connection to the PostgreSQL database
def connect_to_db():
    """Establish a connection to the PostgreSQL database"""
    try:
        connection = psycopg2.connect(
            host=

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