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```python
# AI Dev Tool: FastAPI Pydantic v2 Micro-SaaS with PostgreSQL Vector Search Extension
# ==============================================================================
# This is a production-ready developer code asset for building high-performance
# micro-SaaS applications with FastAPI, Pydantic v2, and PostgreSQL Vector Search Extension.

from fastapi import FastAPI, HTTPException
from fastapi.responses import JSONResponse
from pydantic import BaseModel
from typing import List, Optional
import psycopg2
import psycopg2.extras
import json

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

# Define the Pydantic model for the vector search query
class VectorSearchQuery(BaseModel):
    """Vector search query model"""
    query: str
    limit: Optional[int] = 10

# Define the Pydantic model for the search result
class SearchResult(BaseModel):
    """Search result model"""
    id: int
    title: str
    description: str
    vector: List[float]

# Establish a connection to the PostgreSQL database
def get

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