feat(podcasts): API endpoints — add/list/detail, subscriptions, downloads, sequence, search (task 1.7)

The §8 surface, wired through main.go → router.Config →
/api/podcasts (JWT). Scoping is the engine's (D20): lookups outside the
caller's scope simply return no rows → 404; the only Go-level checks
are explicit-argument authorization (AddFeed's library +
allow_member_feeds gate) and the admin role on feed deletion.

Endpoints: POST/GET /feeds (add: fetch+parse+upsert+subscribe, optional
download_all; list: subscriptions + episode counts + facet values
computed in Go — feed counts are tens), GET /feeds/:id (detail with
download states + gap count), PUT/DELETE /subscription (the ONLY
user-writable podcast state, D9; unsubscribe retains files per D6),
POST /refresh, POST /episodes/:eid/download (+ episode_ids[] bulk and
eid='all' = Download All, exactly the non-downloaded set),
PUT /order (D5 per-user override in the reader_settings KV —
podcast_order:<feed_id>; 'native' deletes the row so default and
explicit choice are indistinguishable), GET /sequence (effective order
— override applied, native fallback — DOWNLOADED-only with the gap
list; this payload powers web chain mode and, per §15, the future
media3 queue), GET /search (subscription-scoped feed + episode search
covering un-downloaded and historical rows, D14), DELETE /feeds/:id
(admin; best-effort file removal, cascade deletes rows).

Gap definition (countGaps): an un-downloaded episode with a downloaded
episode AFTER it in list order — the D8 'middle episodes missing'
banner condition.

internal/router/router.go

@@ -80,6 +80,9 @@ type Config struct {
 	SidecarHandler          *handlers.SidecarHandler
 	ReaderHandler           *handlers.ReaderHandler
 	LibraryService          *services.LibraryService
+	PodcastHandler          *handlers.PodcastHandler
+	PodcastService          *services.PodcastService
+	PodcastWorker           *services.PodcastDownloadWorker
 }
 
 // getBaseURL returns the configured base URL from the database, falling back to
@@ -344,6 +347,7 @@ func RegisterRoutes(cfg *Config) *handlers.Handler {
 	registerConflictRoutes(cfg)
 	registerAnalyticsRoutes(cfg)
 	registerQueueRoutes(cfg)
+	registerPodcastRoutes(cfg)
 	registerJobRoutes(cfg)
 	registerFiltersRoutes(cfg)
 	registerOPDSRoutes(cfg)
@@ -394,6 +398,14 @@ func RegisterRoutes(cfg *Config) *handlers.Handler {
 	// Start background tasks (queue processor and connection cleanup)
 	scannerHandler.StartBackgroundTasks()
 
+	// Podcast subsystem background loops: sequential download worker +
+	// feed poller (plan §7). The poller re-reads its interval setting
+	// every tick; both exit with the server context.
+	if cfg.PodcastWorker != nil && cfg.PodcastService != nil {
+		cfg.PodcastWorker.Start(context.Background())
+		services.NewPodcastPoller(cfg.Queries, cfg.PodcastService, cfg.PodcastWorker, cfg.Settings).Start(context.Background())
+	}
+
 	// Start watch mode for all libraries (after 2 second delay for DB)
 	go func() {
 		time.Sleep(2 * time.Second)