Add production infrastructure: compiled builds, clustering, connection pooling

Root cause of 502 errors under 30 concurrent users: the production server
was running dev-mode infrastructure (Vite dev server, NestJS --watch,
no DB connection pooling, single Node.js process).

Changes:
- backend/Dockerfile: multi-stage prod build (compiled JS, no devDeps)
- frontend/Dockerfile: multi-stage prod build (static assets served by nginx)
- frontend/nginx.conf: SPA routing config for frontend container
- docker-compose.prod.yml: production overlay with tuned Postgres, memory
  limits, health checks, restart policies
- nginx/production.conf: keepalive upstreams, proxy buffering, rate limiting
- backend/src/main.ts: Node.js clustering (1 worker per CPU, up to 4),
  conditional request logging, production CORS
- backend/src/app.module.ts: TypeORM connection pool (max 30, min 5)
- docs/DEPLOYMENT.md: new Production Deployment section

Deploy with: docker compose -f docker-compose.yml -f docker-compose.prod.yml up -d --build

Co-Authored-By: Claude Opus 4.6 <noreply@anthropic.com>

backend/src/app.module.ts

@@ -43,6 +43,13 @@ import { ScheduleModule } from '@nestjs/schedule';
         autoLoadEntities: true,
         synchronize: false,
         logging: false,
+        // Connection pool — reuse connections instead of creating new ones per query
+        extra: {
+          max: 30,                        // max pool size (across all concurrent requests)
+          min: 5,                         // keep at least 5 idle connections warm
+          idleTimeoutMillis: 30000,       // close idle connections after 30s
+          connectionTimeoutMillis: 5000,  // fail fast if pool is exhausted
+        },
       }),
     }),
     DatabaseModule,