feat(desktop): add macOS dock badge for chats with unseen activity

Show a count of chats (root sessions) with unseen activity on the macOS
dock icon. The count is computed in the existing tray snapshot (full
cross-project list, not the capped tray view; a subtask's unseen rolls up
to its root only when subtask notifications are enabled) and pushed to the
main process over the existing desktop_tray_update IPC, which calls
app.setBadgeCount (0 clears it). The badge clears as sessions are marked
seen on window focus.

Add a Dock badge toggle in Appearance settings (default on, persisted,
darwin desktop only), localized across all dictionaries, with a matching
settings-search entry whose availability mirrors the render guard exactly.

packages/ui/src/lib/settings/search.ts

@@ -20,6 +20,8 @@ export interface SettingsSearchResult extends SettingsSearchItem {
 interface SettingsSearchAvailabilityContext extends SettingsRuntimeContext {
   isMobile: boolean;
   isDesktopLocalOrigin: boolean;
+  // macOS desktop shell — for controls that only render on darwin (e.g. dock badge).
+  isMac: boolean;
 }
 
 const SETTINGS_SEARCH_ITEMS: readonly SettingsSearchItem[] = [
@@ -65,6 +67,16 @@ const SETTINGS_SEARCH_ITEMS: readonly SettingsSearchItem[] = [
     keywords: ['transparent', 'transparency', 'vibrancy', 'blur', 'macos', 'opaque'],
     isAvailable: (ctx) => ctx.isDesktopLocalOrigin,
   },
+  {
+    id: 'appearance.dock-badge',
+    page: 'appearance',
+    titleKey: 'settings.openchamber.visual.field.dockBadge',
+    descriptionKey: 'settings.openchamber.visual.field.dockBadgeHint',
+    keywords: ['dock', 'badge', 'unread', 'unseen', 'counter', 'count', 'notification', 'macos'],
+    // Exactly matches the render guard in OpenChamberVisualSettings: any darwin
+    // Electron shell (isMac already implies isDesktopShell), local or remote host.
+    isAvailable: (ctx) => ctx.isMac,
+  },
   {
     id: 'appearance.pwa-install-name',
     page: 'appearance',