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/Services/PodcastLibrary.swift

@@ -0,0 +1,31 @@
+import Foundation
+import Observation
+
+/// Observable, in-memory mirror of `PodcastIndex` so SwiftUI updates immediately
+/// on changes — the Podcasts tab badge and the library list both read from here.
+/// Route podcast mutations through this store so the unplayed badge stays in sync.
+@Observable
+@MainActor
+final class PodcastLibrary {
+    static let shared = PodcastLibrary()
+
+    private(set) var entries: [PodcastEntry] = []
+
+    var unplayed: [PodcastEntry] { entries.filter { !$0.isPlayed } }
+    var played: [PodcastEntry] { entries.filter { $0.isPlayed } }
+    var unplayedCount: Int { entries.reduce(0) { $0 + ($1.isPlayed ? 0 : 1) } }
+
+    private init() { entries = PodcastIndex.all() }
+
+    func reload() { entries = PodcastIndex.all() }
+
+    func setPlayed(_ articleUrl: String, _ played: Bool) {
+        PodcastIndex.setPlayed(articleUrl: articleUrl, played)
+        reload()
+    }
+
+    func remove(_ articleUrl: String) {
+        PodcastIndex.remove(articleUrl: articleUrl)
+        reload()
+    }
+}