⚡ Live Sandbox: AI Dev Tool: TheoremDB A public

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```python
# AI Dev Tool: TheoremDB Vector Search CLI
# ========================================
# A command-line interface for interacting with TheoremDB and performing vector searches.

import os
import sqlite3
import requests
from typing import List, Dict
from fastapi import FastAPI, HTTPException
from fastapi.responses import JSONResponse
from pydantic import BaseModel
from postgresql_vector_search import VectorSearch

# Define the TheoremDB API endpoint
THEOREM_DB_API = "https://api.theoremdb.com"

# Define the PostgreSQL connection settings
POSTGRESQL_HOST = "localhost"
POSTGRESQL_PORT = 5432
POSTGRESQL_USER = "theoremdb"
POSTGRESQL_PASSWORD = "theoremdb"
POSTGRESQL_DB = "theoremdb"

# Define the vector search extension
VECTOR_SEARCH_EXTENSION = "vector_search"

# Create a FastAPI application
app = FastAPI()

# Define a Pydantic model for the search query
class SearchQuery(BaseModel):
    query: str
    limit: int = 10

# Define a function to perform a vector search
def perform_vector_search(query: str, limit: int = 10) -> List[Dict]:
    """
    Perform a vector search on

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