feat(backend): per-tier, per-kind active-game limits with a guest funnel

Cap a player's simultaneous unfinished games per kind (vs_ai, random,
friends) with independent guest and durable-account tiers, held in a new
single-row backend.config table (-1 = unlimited) behind an in-memory cache
and editable live in the admin console (/_gm/limits). Each game is tagged
with games.game_kind on creation.

This replaces the earlier flat MaxActiveQuickGames=10 combined cap: the
per-tier/kind config is the single mechanism, enforced at the same handler
gate (ensureUnderGameLimit by kind on lobby/enqueue) plus the durable
friends cap in CreateInvitation. game.Service.AtGameLimit only resolves the
tier and counts; the limit policy stays at the request edge.

Guests are now refused friend requests, friend-code redemption,
befriend-in-game and invitation creation outright (403 guest_forbidden) --
previously only the UI hid these.

Admin: a kind column in both game lists and the config editor.

Defaults: guest 1 vs_ai / 1 random / 0 friends; durable 10 / 10 / 10.

backend/cmd/backend/main.go

@@ -28,6 +28,7 @@ import (
 	"scrabble/backend/internal/engine"
 	"scrabble/backend/internal/feedback"
 	"scrabble/backend/internal/game"
+	"scrabble/backend/internal/gamelimits"
 	"scrabble/backend/internal/link"
 	"scrabble/backend/internal/lobby"
 	"scrabble/backend/internal/notify"
@@ -156,6 +157,17 @@ func run(ctx context.Context, cfg config.Config, logger *zap.Logger) error {
 	logger.Info("active dictionary version", zap.String("version", games.ActiveVersion()))
 	games.SetNotifier(hub)
 	games.SetMetrics(tel.MeterProvider().Meter("scrabble/backend/game"))
+
+	// Active-game limit config: the per-tier, per-kind caps in backend.config, read once into an
+	// in-memory cache at boot and refreshed when the admin edits them. A boot-time load fails fast if
+	// the single config row is missing; the game domain reads the cache on every new-game gate.
+	gameLimits := gamelimits.NewService(gamelimits.NewStore(db))
+	if err := gameLimits.Load(ctx); err != nil {
+		return fmt.Errorf("load game-limit config: %w", err)
+	}
+	games.SetGameLimits(gameLimits)
+	logger.Info("game-limit config loaded")
+
 	go games.RunSweeper(ctx, cfg.Game.TimeoutSweepInterval)
 	logger.Info("game turn-timeout sweeper started",
 		zap.Duration("interval", cfg.Game.TimeoutSweepInterval))
@@ -305,6 +317,7 @@ func run(ctx context.Context, cfg config.Config, logger *zap.Logger) error {
 		BanView:       banView,
 		Ads:           adsSvc,
 		Payments:      paymentsSvc,
+		GameLimits:    gameLimits,
 		Notifier:      hub,
 		ExportSignKey: cfg.ExportSignKey,
 		Renderer:      renderer,