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/internal/game/service.go

@@ -54,8 +54,14 @@ type Service struct {
 	// have committed a move (a nudge answered by moving stops counting as unread). It is
 	// best-effort and kept as a func so the game package never imports the social package.
 	clearNudges func(ctx context.Context, gameID, accountID uuid.UUID) error
-	metrics     *gameMetrics
-	log         *zap.Logger
+	// expireNudges, when set, marks every pending nudge in a game read once the game
+	// finishes (the nudge badge is stale on a completed game). Unlike clearNudges it is
+	// keyed by game alone — it clears all seats' nudges, not one mover's — and runs on
+	// every completion path through commit. Best-effort; a func so the game package never
+	// imports the social package.
+	expireNudges func(ctx context.Context, gameID uuid.UUID) error
+	metrics      *gameMetrics
+	log          *zap.Logger
 }
 
 // NewService constructs a Service. store and accounts wrap the same pool;
@@ -107,6 +113,15 @@ func (svc *Service) SetNudgeClearer(fn func(ctx context.Context, gameID, account
 	svc.clearNudges = fn
 }
 
+// SetNudgeExpirer installs the hook that marks every pending nudge in a game read once the
+// game finishes, on any completion path (a closing move, a resignation, a turn-timeout or a
+// forfeit). It must be called during startup wiring; the default (nil) leaves a finished
+// game's nudges to expire only when a recipient opens the move history or chat. The social
+// package wires its ExpireNudges here. Chat messages are deliberately left unread.
+func (svc *Service) SetNudgeExpirer(fn func(ctx context.Context, gameID uuid.UUID) error) {
+	svc.expireNudges = fn
+}
+
 // SetFirstMoveEntropy overrides the entropy source for the first-move draw
 // (docs/ARCHITECTURE.md §6). It must be called during wiring or test setup before any
 // game is created; the production default is crypto/rand and is never overridden.
@@ -699,6 +714,16 @@ func (svc *Service) commit(ctx context.Context, gameID uuid.UUID, g *engine.Game
 	}
 	if c.finished {
 		svc.cache.remove(gameID)
+		// A finished game's nudges are stale, so clear them all here — every completion path
+		// funnels through commit (a closing move, a resignation, a forfeit or a turn-timeout),
+		// and only the move path also clears the mover's nudge on its own. Best-effort like
+		// clearNudges: the finish has committed, so a cleanup failure is logged, not surfaced.
+		// ExpireNudges leaves chat messages unread.
+		if svc.expireNudges != nil {
+			if err := svc.expireNudges(ctx, gameID); err != nil {
+				svc.log.Warn("expire nudges on game finish", zap.Error(err))
+			}
+		}
 	}
 	post, err := svc.store.GetGame(ctx, gameID)
 	if err != nil {
@@ -1440,13 +1465,24 @@ func (svc *Service) voidGame(ctx context.Context, pre Game, g *engine.Game) erro
 	if err != nil {
 		return err
 	}
-	return svc.store.VoidGame(ctx, voidCommit{
+	if err := svc.store.VoidGame(ctx, voidCommit{
 		gameID:    pre.ID,
 		endReason: g.Reason().String(),
 		scores:    scores,
 		now:       svc.clock(),
 		stats:     buildStats(g, statSeats),
-	})
+	}); err != nil {
+		return err
+	}
+	// A voided game is finished (as a draw) but bypasses commit, so clear its now-stale nudges
+	// here too. Best-effort, like the commit path: the void has persisted, so a cleanup failure
+	// is logged, not surfaced.
+	if svc.expireNudges != nil {
+		if err := svc.expireNudges(ctx, pre.ID); err != nil {
+			svc.log.Warn("expire nudges on voided game", zap.Error(err))
+		}
+	}
+	return nil
 }
 
 // replayMove re-applies one journalled move to g through the decoded engine API.