feat(podcasts): #6 quick wins — tab badge, remember speed, sleep timer, persist queue

- Unread badge on the Podcasts tab via a new observable PodcastLibrary store
  that mirrors PodcastIndex; all played/remove/generation mutations route
  through it so the count stays live.
- Persist playback speed across sessions (UserDefaults), applied on setup.
- Sleep timer (15/30/45 min or end-of-episode) with a moon menu in the full
  player; end-of-episode stops instead of auto-advancing.
- Persist the current episode + Up Next queue and restore a *paused* session on
  launch; save position when backgrounding. Audio session is only claimed on
  actual playback so a restored session never interrupts other audio.

Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>

Marks/MarksApp.swift

@@ -38,6 +38,7 @@ struct MainContainer: View {
     @State private var showDeepLinkPlayer = false
     @State private var selectedTab: AppTab = .bookmarks
     @State private var router = IntentRouter.shared
+    @State private var library = PodcastLibrary.shared
     @Environment(\.scenePhase) private var scenePhase
 
     init(config: ServerConfig, onDisconnect: @escaping () -> Void) {
@@ -58,6 +59,7 @@ struct MainContainer: View {
             Tab("Podcasts", systemImage: "headphones", value: AppTab.podcasts) {
                 PodcastLibraryView(vm: viewModel.podcastPlayer, claude: viewModel.claude)
             }
+            .badge(library.unplayedCount)
             Tab(value: AppTab.search, role: .search) {
                 SearchView(viewModel: viewModel)
             }
@@ -65,7 +67,11 @@ struct MainContainer: View {
         .onOpenURL { url in
             handleDeepLink(url)
         }
-        .task { applyPendingIntent(); viewModel.processPendingPodcastRequests() }
+        .task {
+            applyPendingIntent()
+            viewModel.processPendingPodcastRequests()
+            viewModel.podcastPlayer.restoreSession(claude: viewModel.claude)
+        }
         .onChange(of: router.openBookmarkURL) { _, _ in applyPendingIntent() }
         .onChange(of: router.searchRequest) { _, _ in applyPendingIntent() }
         .onChange(of: scenePhase) { old, new in
@@ -75,6 +81,11 @@ struct MainContainer: View {
                 viewModel.processPendingPodcastRequests()
                 Task { await viewModel.load() }
             }
+            // Persist playback position + session when leaving the foreground so a
+            // relaunch can resume where you left off.
+            if new == .background {
+                viewModel.podcastPlayer.saveProgress()
+            }
         }
         .sheet(item: $deepLinkBrowser) { item in
             BrowserView(url: item.url, title: item.url.host ?? "", claude: viewModel.claude, podcastPlayer: viewModel.podcastPlayer, podcastGenerator: viewModel.podcastGenerator)