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```python
# AI Dev Tool: FastAPI Pydantic v2 Micro-SaaS Boilerplate with PostgreSQL Vector Search Extension
#============================================================================================
from fastapi import FastAPI, HTTPException
from pydantic import BaseModel
from typing import List, Optional
import psycopg2
from psycopg2 import Error
from psycopg2.extras import DictCursor
import json

# Define the FastAPI application
app = FastAPI()

# Define the PostgreSQL connection parameters
POSTGRES_HOST = "localhost"
POSTGRES_DB = "mydatabase"
POSTGRES_USER = "myuser"
POSTGRES_PASSWORD = "mypassword"

# Define the PostgreSQL connection function
def connect_to_postgres():
    try:
        connection = psycopg2.connect(
            host=POSTGRES_HOST,
            database=POSTGRES_DB,
            user=POSTGRES_USER,
            password=POSTGRES_PASSWORD,
            cursor_factory=DictCursor
        )
        return connection
    except Error as e:
        print(f"Error connecting to PostgreSQL: {e}")
        raise HTTPException(status_code=500, detail="Failed to connect to PostgreSQL")

# Define the Vector Search model
class VectorSearchModel(BaseModel):
    query: str
    limit: Optional[int] = 10

# Define the

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