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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
# with FastAPI, Pydantic v2, and 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 logging

# Initialize the logger
logging.basicConfig(level=logging.INFO)
logger = logging.getLogger(__name__)

# Define the FastAPI application
app = FastAPI(title="FastAPI Pydantic v2 PostgreSQL Vector Search Extension",
              description="A micro-SaaS application with FastAPI, Pydantic v2, and PostgreSQL Vector Search Extension.",
              version="1.0.0")

# Define the Pydantic model for the search query
class SearchQuery(BaseModel):
    """The search query model."""
    query: str
    limit: Optional[int] = 10

# Define the Pydantic model for the search result
class SearchResult(BaseModel):
    """The search result model."""
    id: int
    title: str
    description: str

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