Add backend health check, fetch timeouts, stale token cleanup, and error screen

Frontend now checks /health before starting auth flow. Shows a clear
"Cannot Connect to Server" screen with retry button when backend is
unreachable. Stale non-JWT tokens in localStorage are detected and
cleared automatically. All API calls have a 10s timeout via AbortController.

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

apps/web/src/App.tsx

@@ -7,7 +7,7 @@ import { InviteGateScreen } from '@/components/game/InviteGateScreen';
 import { useGameLoop } from '@/hooks/useGameLoop';
 import { useAuthGate } from '@/hooks/useAuthGate';
 import { TICK_INTERVAL_MS } from '@ai-tycoon/shared';
-import { Sparkles } from 'lucide-react';
+import { Sparkles, RefreshCw, WifiOff } from 'lucide-react';
 
 function LoadingScreen() {
   return (
@@ -25,8 +25,35 @@ function LoadingScreen() {
   );
 }
 
+function BackendErrorScreen({ error, onRetry }: { error: string; onRetry: () => void }) {
+  return (
+    <div className="min-h-screen flex items-center justify-center bg-gradient-to-b from-surface-950 to-surface-900">
+      <div className="max-w-md w-full mx-4 text-center">
+        <div className="inline-flex items-center gap-2 mb-6">
+          <Sparkles className="text-accent-light" size={32} />
+          <h1 className="text-4xl font-bold bg-gradient-to-r from-accent-light to-accent bg-clip-text text-transparent">
+            AI Tycoon
+          </h1>
+        </div>
+
+        <div className="bg-surface-900 border border-surface-700 rounded-xl p-6 space-y-4">
+          <WifiOff className="mx-auto text-danger" size={40} />
+          <h2 className="text-lg font-semibold text-surface-100">Cannot Connect to Server</h2>
+          <p className="text-sm text-surface-400">{error}</p>
+          <button
+            onClick={onRetry}
+            className="inline-flex items-center gap-2 px-5 py-2.5 rounded-lg bg-accent hover:bg-accent-dark text-white font-medium text-sm transition-colors"
+          >
+            <RefreshCw size={16} /> Retry
+          </button>
+        </div>
+      </div>
+    </div>
+  );
+}
+
 export function App() {
-  const { loading: authLoading, needsInvite, needsPasswordReset, setRegistered, setNeedsPasswordReset } = useAuthGate();
+  const { loading: authLoading, backendError, needsInvite, needsPasswordReset, setRegistered, setNeedsPasswordReset, retry } = useAuthGate();
   const companyName = useGameStore((s) => s.meta.companyName);
   const lastTickTimestamp = useGameStore((s) => s.meta.lastTickTimestamp);
   const [catchUpTicks, setCatchUpTicks] = useState<number | null>(null);
@@ -43,12 +70,16 @@ export function App() {
     }
   }, [companyName, lastTickTimestamp, catchUpDone]);
 
-  useGameLoop(!catchUpDone || authLoading || needsInvite || needsPasswordReset);
+  useGameLoop(!catchUpDone || authLoading || !!backendError || needsInvite || needsPasswordReset);
 
   if (authLoading) {
     return <LoadingScreen />;
   }
 
+  if (backendError) {
+    return <BackendErrorScreen error={backendError} onRetry={retry} />;
+  }
+
   if (needsInvite || needsPasswordReset) {
     return (
       <InviteGateScreen