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```python
# AI Dev Tool: PostgreSQL Vector Search Extension with FastAPI and Supabase
# ========================================================================
# This is a production-ready developer code asset for building AI-native SaaS applications
# with PostgreSQL Vector Search Extension, FastAPI, and Supabase.

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

# Initialize Supabase client
SUPABASE_URL = os.environ.get("SUPABASE_URL")
SUPABASE_KEY = os.environ.get("SUPABASE_KEY")
supabase: Client = create_client(SUPABASE_URL, SUPABASE_KEY)

# Initialize FastAPI app
app = FastAPI()

# Define data models
class VectorSearchQuery(BaseModel):
    """Vector search query model"""
    query: str
    limit: int = 10

class VectorSearchResult(BaseModel):
    """Vector search result model"""
    id: int
    vector: List[float]
    score: float

# Define API endpoints
@app.post("/vector_search")
async def vector_search(query: VectorSearchQuery):
    """
    Perform vector search on the PostgreSQL

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