Show unread on the installed app icon #258

Parent

#249 Deepen the installed PWA.

What to build

If navigator.setAppBadge exists, mirror the header-bell unread count. That store field is totalNotificationUnreadCount (notification unread). It is not TOC chat unread and not a sum of rooms.

Clear the badge at zero or when signed out.

Acceptance criteria

• Installed app with the Badging API: the icon count matches totalNotificationUnreadCount.
• Zero unread or sign-out: clearAppBadge.
• Only the signed-in page sets or clears the badge. The push service worker does not write it.
• Home does not grow a notification fetch for this. The badge updates when a surface that already runs useNotificationCount is open (the pad).
• Missing API: no-op. No toast. In-app UnreadBadge stays.

Out of scope

• A second unread source
• Summing chat-channel unread
• Badge while signed out
• A native dock menu