fix(game): clear nudges on game completion

Nudge badges lingered in the lobby on games that ended by turn-timeout,
resignation or forfeit: those paths commit the finish directly, bypassing
the move path's per-mover NudgeClearer, so the awaited seat's nudge was
never marked read. A finished game's nudges are stale, so clear them all.

Add a wired NudgeExpirer (social.ExpireNudges) called from the shared
commit finish block — covering every completion path — and from the
voidGame recovery path. It clears every seat's nudge bits for the game
and leaves chat messages unread; unlike ClearNudges it records no
publish-to-read latency, since a completion is an expiry, not a read.

An integration test reproduces the timeout case (nudge cleared, chat
kept). Docs: ARCHITECTURE §9.1, FUNCTIONAL (+_ru), backend/README.

backend/cmd/backend/main.go

@@ -180,8 +180,10 @@ func run(ctx context.Context, cfg config.Config, logger *zap.Logger) error {
 	socialSvc := social.NewService(social.NewStore(db), accounts, games)
 	socialSvc.SetNotifier(hub)
 	socialSvc.SetMetrics(tel.MeterProvider().Meter("scrabble/backend/social"))
-	// A nudge the recipient answered by moving is marked read on the move path.
+	// A nudge the recipient answered by moving is marked read on the move path; every nudge in a
+	// game is marked read when the game finishes (a stale badge), on any completion path.
 	games.SetNudgeClearer(socialSvc.ClearNudges)
+	games.SetNudgeExpirer(socialSvc.ExpireNudges)
 	// Reap per-game disguised-robot friend requests once their game is long finished
 	// (the robot ignores them; the row only pins the in-game "request sent" state).
 	robotReqReaper := social.NewRobotFriendRequestReaper(socialSvc, logger)