feat(pwa): app-icon badge with the highlight count via navigator.setAppBadge (#154)

Same number as the tab title (leaf-room highlights), cleared at zero.
Skipped under Tauri where the native set_badge_count owns the badge, and
silently absent where the Badging API is not available.

Closes #154

Co-Authored-By: Claude Opus 5 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_01PPmy3tPq869XDW4njjVaKA

src/app/pages/client/ClientNonUIFeatures.tsx

@@ -59,7 +59,7 @@ import { toastQueueAtom } from '../../state/toast';
 import { useReminders } from '../../hooks/useReminders';
 import { getRoomRetentionMs, isExpired } from '../../utils/retention';
 import { useTauriUpdater } from '../../hooks/useTauriUpdater';
-import { invokeTauri } from '../../hooks/useTauri';
+import { invokeTauri, isTauri as isTauriApp } from '../../hooks/useTauri';
 import { TauriDesktopFeatures } from '../../components/TauriDesktopFeatures';
 import { KeyboardShortcutsDialog, useKeyboardShortcutsTrigger } from '../../features/shortcuts';
 import { useRoomsListener } from '../../hooks/useRoomsListener';
@@ -153,6 +153,23 @@ function FaviconUpdater() {
     } else {
       document.title = 'Lotus Chat';
     }
+
+    // [Gitea #154] Installed-PWA app-icon badge (Android Chrome, desktop
+    // Chrome/Edge). Same number as the title so the two never disagree:
+    // highlights only, mirroring the desktop app's native `set_badge_count`
+    // (which owns the badge under Tauri, so skip there). Plain tabs and
+    // browsers without the Badging API simply have no `setAppBadge`.
+    if (!isTauriApp()) {
+      const nav = navigator as Navigator & {
+        setAppBadge?: (count?: number) => Promise<void>;
+        clearAppBadge?: () => Promise<void>;
+      };
+      if (totalHighlight > 0) {
+        nav.setAppBadge?.(totalHighlight).catch(() => {});
+      } else {
+        nav.clearAppBadge?.().catch(() => {});
+      }
+    }
   }, [roomToUnread]);
 
   return null;