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```python
# AI Dev Tool: BrainSync - FastAPI + PostgreSQL Vector Search Extension
# Description: Micro-SaaS boilerplate for brain-inspired vector search API

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

# PostgreSQL connection settings
POSTGRES_HOST = "localhost"
POSTGRES_DB = "brain_sync"
POSTGRES_USER = "brain_sync_user"
POSTGRES_PASSWORD = "brain_sync_password"

# Create a new FastAPI app
app = FastAPI()

# Define a Pydantic model for vector search queries
class VectorSearchQuery(BaseModel):
    """Vector search query model"""
    query_vector: List[float]
    limit: Optional[int] = 10

# Define a Pydantic model for search results
class SearchResult(BaseModel):
    """Search result model"""
    id: int
    vector: List[float]
    score: float

# Establish a PostgreSQL connection
def get_db_connection():
    """Get a PostgreSQL database connection"""
    try:
        return psycopg2.connect(
            host=POSTGRES_HOST,
            database=POSTGRES_DB

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