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```python
# AI Dev Tool: PostgreSQL Vector Search Extension with FastAPI and Supabase
# ==============================================================================
# This is a high-demand, production-ready developer code asset for building
# scalable and efficient vector search applications using PostgreSQL, FastAPI,
# and Supabase.

import os
import uvicorn
from fastapi import FastAPI, HTTPException
from fastapi.responses import JSONResponse
from fastapi.requests import Request
from pydantic import BaseModel
from supabase import create_client, Client
from typing import List, Dict

# Supabase Configuration
SUPABASE_URL = os.environ.get("SUPABASE_URL")
SUPABASE_KEY = os.environ.get("SUPABASE_KEY")
SUPABASE_SECRET = os.environ.get("SUPABASE_SECRET")

# Create a Supabase client
supabase: Client = create_client(SUPABASE_URL, SUPABASE_KEY)

# Define the FastAPI application
app = FastAPI()

# Define a Pydantic model for vector search queries
class VectorSearchQuery(BaseModel):
    """Vector search query model"""
    query: str
    limit: int = 10

# Define a Pydantic model for vector search results
class VectorSearchResult(BaseModel):
    """Vector search

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