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/internal/server/handlers_admin_limits.go

@@ -0,0 +1,65 @@
+package server
+
+import (
+	"fmt"
+
+	"github.com/gin-gonic/gin"
+
+	"scrabble/backend/internal/adminconsole"
+	"scrabble/backend/internal/gamelimits"
+)
+
+// consoleLimits renders the per-tier, per-kind active-game limit form (backend.config), read from
+// the in-memory cache.
+func (s *Server) consoleLimits(c *gin.Context) {
+	cfg := s.gamelimits.Get()
+	s.renderConsole(c, "limits", "limits", "Game limits", adminconsole.GameLimitsView{
+		GuestVsAI:      cfg.Guest.VsAI,
+		GuestRandom:    cfg.Guest.Random,
+		GuestFriends:   cfg.Guest.Friends,
+		DurableVsAI:    cfg.Durable.VsAI,
+		DurableRandom:  cfg.Durable.Random,
+		DurableFriends: cfg.Durable.Friends,
+	})
+}
+
+// consoleUpdateLimits saves the active-game limits and refreshes the hot cache in place. Each
+// field is a per-kind cap: -1 is unlimited, 0 blocks the kind, a positive value caps concurrent games.
+func (s *Server) consoleUpdateLimits(c *gin.Context) {
+	cfg := gamelimits.Config{
+		Guest: gamelimits.Limits{
+			VsAI:    atoiForm(c, "guest_vs_ai"),
+			Random:  atoiForm(c, "guest_random"),
+			Friends: atoiForm(c, "guest_friends"),
+		},
+		Durable: gamelimits.Limits{
+			VsAI:    atoiForm(c, "durable_vs_ai"),
+			Random:  atoiForm(c, "durable_random"),
+			Friends: atoiForm(c, "durable_friends"),
+		},
+	}
+	if err := validateGameLimits(cfg); err != nil {
+		s.renderConsoleMessage(c, "Invalid", err.Error(), "/_gm/limits")
+		return
+	}
+	if err := s.gamelimits.Update(c.Request.Context(), cfg); err != nil {
+		s.consoleError(c, err)
+		return
+	}
+	s.renderConsoleMessage(c, "Saved", "game limits updated", "/_gm/limits")
+}
+
+// validateGameLimits rejects a limit below -1 (the unlimited sentinel); -1, 0 and any positive count
+// are valid. It mirrors the backend.config CHECK so a bad value is refused with a clean message
+// rather than a raw database error.
+func validateGameLimits(cfg gamelimits.Config) error {
+	for _, v := range []int{
+		cfg.Guest.VsAI, cfg.Guest.Random, cfg.Guest.Friends,
+		cfg.Durable.VsAI, cfg.Durable.Random, cfg.Durable.Friends,
+	} {
+		if v < gamelimits.Unlimited {
+			return fmt.Errorf("a limit must be -1 (unlimited), 0, or a positive count")
+		}
+	}
+	return nil
+}