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/Dockerfile

@@ -0,0 +1,26 @@
+# ---- Production Dockerfile for NestJS backend ----
+# Multi-stage build: compile TypeScript, then run with minimal image
+
+# Stage 1: Build
+FROM node:20-alpine AS builder
+WORKDIR /app
+COPY package*.json ./
+RUN npm ci
+COPY . .
+RUN npm run build
+
+# Stage 2: Production
+FROM node:20-alpine
+WORKDIR /app
+
+# Only install production dependencies
+COPY package*.json ./
+RUN npm ci --omit=dev && npm cache clean --force
+
+# Copy compiled output from builder
+COPY --from=builder /app/dist ./dist
+
+EXPOSE 3000
+
+# Run the compiled JS directly — no ts-node, no watch, no devDeps
+CMD ["node", "dist/main"]