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/inttest/chatread_test.go

@@ -138,6 +138,65 @@ func TestNudgeClearedByMove(t *testing.T) {
 	}
 }
 
+// TestGameCompletionExpiresNudgesKeepsChat checks that finishing a game by turn-timeout marks every
+// pending nudge in it read — the lobby's nudge badge is stale once the game is over — while leaving
+// real chat messages unread. It reproduces the reported bug: the timeout path commits the finish
+// directly, bypassing the move path's per-mover nudge clear, so without a completion-driven expiry
+// the awaited seat's nudge lingered as a badge on the finished game.
+func TestGameCompletionExpiresNudgesKeepsChat(t *testing.T) {
+	ctx := context.Background()
+	gameSvc := newGameService()
+	socialSvc := newSocialService()
+	gameSvc.SetNudgeExpirer(socialSvc.ExpireNudges)
+
+	seats := []uuid.UUID{provisionAccount(t), provisionAccount(t)}
+	g, err := gameSvc.Create(ctx, game.CreateParams{
+		Variant: engine.VariantEnglish, Seats: seats, TurnTimeout: time.Hour, Seed: openingSeed(t),
+	})
+	if err != nil {
+		t.Fatalf("create: %v", err)
+	}
+
+	// Seat 1 nudges the to-move seat 0 (the awaited player), and seat 0 posts a real chat message,
+	// which seat 1 then holds unread. So before the timeout each side has exactly one unread entry:
+	// seat 0 a nudge, seat 1 a message — letting HasUnread isolate each kind.
+	if _, err := socialSvc.Nudge(ctx, g.ID, seats[1]); err != nil {
+		t.Fatalf("nudge: %v", err)
+	}
+	if _, err := socialSvc.PostMessage(ctx, g.ID, seats[0], "good luck", ""); err != nil {
+		t.Fatalf("post message: %v", err)
+	}
+	if unread, _ := socialSvc.HasUnread(ctx, g.ID, seats[0]); !unread {
+		t.Fatal("seat 0 should hold the nudge unread before the timeout")
+	}
+	if unread, _ := socialSvc.HasUnread(ctx, g.ID, seats[1]); !unread {
+		t.Fatal("seat 1 should hold the message unread before the timeout")
+	}
+
+	// Age the turn past its deadline and time seat 0 out; an empty away window keeps this
+	// deterministic regardless of the wall clock. The sweep finishes the game through the direct
+	// commit path, never the move path.
+	backdate(t, g.ID, time.Now().UTC().Add(-2*time.Hour))
+	setAway(t, seats[0], "UTC", "00:00", "00:00")
+	if n, err := gameSvc.SweepTimeouts(ctx, time.Now().UTC()); err != nil || n < 1 {
+		t.Fatalf("sweep swept %d (err %v), want >= 1", n, err)
+	}
+	if status, reason := gameStatus(t, gameSvc, g.ID); status != game.StatusFinished || reason != "timeout" {
+		t.Fatalf("game not timed out: status %q reason %q", status, reason)
+	}
+
+	// The nudge is stale on a finished game and must be cleared; the chat message must survive.
+	if unread, _ := socialSvc.HasUnread(ctx, g.ID, seats[0]); unread {
+		t.Error("the nudge should be expired once the game has finished")
+	}
+	if unread, _ := socialSvc.HasUnread(ctx, g.ID, seats[1]); !unread {
+		t.Error("a real chat message must stay unread after the game finishes")
+	}
+	if msg, _ := socialSvc.HasUnreadMessage(ctx, g.ID, seats[1]); !msg {
+		t.Error("the chat message must remain flagged as an unread message after completion")
+	}
+}
+
 // TestChatToRobotIsBornRead checks a text message to a disguised robot opponent (a pooled
 // robot substituted into an ordinary, non-AI game) is born read: the robot never opens the
 // chat, so the message must not linger unread (skewing the count and the read metric).