feat: Windows taskbar notification badge counter

Adds a Tauri command (set_badge_count) that draws a red circle badge
with a highlight count onto the Windows taskbar button overlay icon,
matching Discord's behavior. Badge shows @mention/highlight count only
(not total messages), clears to zero when all highlights are read.

Frontend: useTauriNotificationBadge hook reads roomToUnreadAtom and
calls set_badge_count via window.__TAURI_INTERNALS__.invoke whenever
the unread map changes. No-ops silently in the browser (non-Tauri).
Mounted as TauriEffects inside JotaiProvider in App.tsx.

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

src/app/hooks/useTauriNotificationBadge.ts

@@ -0,0 +1,25 @@
+import { useEffect } from 'react';
+import { useAtomValue } from 'jotai';
+import { roomToUnreadAtom } from '../state/room/roomToUnread';
+
+// Tauri v2 injects __TAURI_INTERNALS__ into the webview at runtime.
+// We use it directly so cinny doesn't need @tauri-apps/api as a dependency.
+type TauriInternals = { invoke: (cmd: string, args?: Record<string, unknown>) => Promise<unknown> };
+const tauriInvoke = (): TauriInternals['invoke'] | undefined =>
+  (window as unknown as { __TAURI_INTERNALS__?: TauriInternals }).__TAURI_INTERNALS__?.invoke;
+
+export function useTauriNotificationBadge() {
+  const roomToUnread = useAtomValue(roomToUnreadAtom);
+
+  useEffect(() => {
+    const invoke = tauriInvoke();
+    if (!invoke) return;
+
+    let totalHighlights = 0;
+    roomToUnread.forEach((unread) => {
+      totalHighlights += unread.highlight;
+    });
+
+    invoke('set_badge_count', { count: totalHighlights }).catch(() => {});
+  }, [roomToUnread]);
+}

src/app/pages/App.tsx

@@ -15,6 +15,7 @@ import { ScreenSizeProvider, useScreenSize } from '../hooks/useScreenSize';
 import { useCompositionEndTracking } from '../hooks/useComposingCheck';
 import { settingsAtom } from '../state/settings';
 import { LotusToastContainer } from '../features/toast/LotusToastContainer';
+import { useTauriNotificationBadge } from '../hooks/useTauriNotificationBadge';
 
 const FONT_MAP: Record<string, string> = {
   system: "system-ui, -apple-system, BlinkMacSystemFont, 'Segoe UI', sans-serif",
@@ -52,6 +53,11 @@ function AppearanceEffects() {
   return null;
 }
 
+function TauriEffects() {
+  useTauriNotificationBadge();
+  return null;
+}
+
 function NightLightOverlay() {
   const settings = useAtomValue(settingsAtom);
   if (!settings.nightLightEnabled) return null;
@@ -131,6 +137,7 @@ function App() {
                       <QueryClientProvider client={queryClient}>
                         <JotaiProvider>
                           <AppearanceEffects />
+                          <TauriEffects />
                           <RouterProvider router={createRouter(clientConfig, screenSize)} />
                           <NightLightOverlay />
                           <LotusToastContainer />