feat(web): scaffold Next.js 16 dashboard with design system and auth client

Build out apps/web with Tailwind CSS v4, BetterAuth client, and the Mosaic
design system. Populate @mosaic/design-tokens with actual color palette
(deep blue-grays, blue/purple/teal accents), font definitions (Outfit/Fira Code),
and spacing scale per AD-13.

- Tailwind v4 CSS-first config with @theme tokens in globals.css
- next/font for self-hosted Outfit + Fira Code (no Google CDN)
- App shell with sidebar nav, topbar, and content area
- Route groups: (dashboard) with AppShell, (auth) with centered card
- Placeholder pages: /chat, /login, /register
- BetterAuth React client (useSession, signIn, signUp, signOut)
- Gateway API fetch wrapper with credentials
- cn() utility with clsx + tailwind-merge
- Dark theme default, WCAG-friendly contrast

Refs #26

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

apps/web/package.json

@@ -11,11 +11,16 @@
     "start": "next start"
   },
   "dependencies": {
+    "@mosaic/design-tokens": "workspace:^",
+    "better-auth": "^1.5.5",
+    "clsx": "^2.1.0",
     "next": "^16.0.0",
     "react": "^19.0.0",
-    "react-dom": "^19.0.0"
+    "react-dom": "^19.0.0",
+    "tailwind-merge": "^3.5.0"
   },
   "devDependencies": {
+    "@tailwindcss/postcss": "^4.0.0",
     "@types/node": "^22.0.0",
     "@types/react": "^19.0.0",
     "@types/react-dom": "^19.0.0",