fix(app): say it out loud when wake-ups do not work

Registration failures from the distributor were swallowed, so push looked
enabled while nothing arrived. Without wake-ups Android suspends the app
and messages only show up once the user opens it — which looks like broken
delivery even though the relay has them queued.

The failure is now remembered and surfaced in Settings, and the chat list
carries a dismissible line explaining the state.

app/lib/providers/push_provider.dart

@@ -1,6 +1,7 @@
 import 'package:flutter_riverpod/flutter_riverpod.dart';
 import 'package:sepot/core/push/push_service.dart';
 import 'package:sepot/providers/messaging_provider.dart';
+import 'package:sepot/providers/storage_provider.dart';
 
 /// Push vrstva nad messagingem.
 final pushServiceProvider = Provider<PushService>((ref) {
@@ -18,3 +19,31 @@ final pushInitProvider = FutureProvider<void>((ref) async {
   // re-registruje u relaye po každém startu.
   await push.restoreMode();
 });
+
+/// Stav budíčků pro chat list. Bez funkčního pushe appka na pozadí spí a
+/// zprávy dorazí, až ji uživatel otevře — o tom má vědět dřív, než mu
+/// někdo vyčte, že neodpovídá.
+enum PushHealth {
+  /// Budíčky nastavené a bez chyby.
+  ok,
+
+  /// Push vůbec nezapnutý (výchozí stav po instalaci).
+  off,
+
+  /// Distributor registraci odmítl — zapnuto, ale nefunguje.
+  failed,
+}
+
+final pushHealthProvider = FutureProvider<PushHealth>((ref) async {
+  await ref.watch(pushInitProvider.future);
+  final push = ref.read(pushServiceProvider);
+  if (await push.registrationFailed()) return PushHealth.failed;
+  return await push.currentMode() == 'off' ? PushHealth.off : PushHealth.ok;
+});
+
+/// Skryl si uživatel upozornění na chat listu? Rozhodnutí se respektuje —
+/// připomínat donekonečna by bylo otravné.
+final pushHintDismissedProvider = FutureProvider<bool>((ref) async {
+  final storage = ref.watch(storageProvider);
+  return await storage.getSetting<bool>('push_hint_dismissed') ?? false;
+});