feat(web): wire auth pages with BetterAuth client and route guards

Connect login/register forms to BetterAuth signIn.email() and signUp.email()
with error display, loading states, and disabled inputs during submission.
Add AuthGuard (redirects unauthenticated users to /login) on dashboard routes
and GuestGuard (redirects authenticated users to /chat) on auth routes.
Sign out in topbar now redirects to /login.

Refs #27

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

apps/web/src/components/layout/topbar.tsx

@@ -1,9 +1,16 @@
 'use client';
 
+import { useRouter } from 'next/navigation';
 import { useSession, signOut } from '@/lib/auth-client';
 
 export function Topbar(): React.ReactElement {
   const { data: session } = useSession();
+  const router = useRouter();
+
+  async function handleSignOut(): Promise<void> {
+    await signOut();
+    router.replace('/login');
+  }
 
   return (
     <header className="sticky top-0 z-20 flex h-14 items-center justify-between border-b border-surface-border bg-surface-card/80 px-6 backdrop-blur-sm">
@@ -17,7 +24,7 @@ export function Topbar(): React.ReactElement {
             </span>
             <button
               type="button"
-              onClick={() => signOut()}
+              onClick={handleSignOut}
               className="rounded-md px-3 py-1.5 text-sm text-text-muted transition-colors hover:bg-surface-elevated hover:text-text-primary"
             >
               Sign out