fix AGENTS.md stack ref, add CORS plugin, add .env.example, remove dead index.ts

backend/src/app.ts

@@ -1,4 +1,5 @@
 import Fastify from 'fastify';
+import cors from '@fastify/cors';
 import { Client as PgClient } from 'pg';
 import { createClient as createRedisClient } from 'redis';
 import { DbService } from './services/db.service.js';
@@ -28,6 +29,10 @@ export interface AppConfig {
 export async function buildApp(config: AppConfig) {
   const fastify = Fastify({ logger: true });
 
+  await fastify.register(cors, {
+    origin: process.env.CORS_ORIGIN?.split(',') || ['http://localhost:5173', 'http://localhost:5174'],
+  });
+
   const pgClient = new PgClient({
     connectionString: process.env.DATABASE_URL,
   });

backend/src/index.ts

@@ -1 +0,0 @@
-console.log('Backend starting...');