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/game/service.go

@@ -17,6 +17,7 @@ import (
 
 	"scrabble/backend/internal/account"
 	"scrabble/backend/internal/engine"
+	"scrabble/backend/internal/gamelimits"
 	"scrabble/backend/internal/notify"
 	"scrabble/backend/internal/payments"
 	"scrabble/backend/internal/session"
@@ -57,6 +58,9 @@ type Service struct {
 	// nil, only the free per-game allowance is served (no purchased hints). vs_ai hints are
 	// wallet-free and never touch it.
 	hintWallet HintWallet
+	// limits is the per-tier active-game cap config (cached). Set by SetGameLimits during wiring;
+	// when nil, no active-game limit is enforced (a game is always creatable).
+	limits *gamelimits.Service
 	// clearNudges, when set, marks the actor's pending nudges in a game read once they
 	// 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.
@@ -126,6 +130,37 @@ func (svc *Service) SetHintWallet(w HintWallet) {
 	svc.hintWallet = w
 }
 
+// SetGameLimits installs the active-game limit config (cached), enabling the per-tier caps. When
+// unset (nil), a game is always creatable.
+func (svc *Service) SetGameLimits(l *gamelimits.Service) {
+	svc.limits = l
+}
+
+// AtGameLimit reports whether accountID has reached its tier's active-game cap for kind — the
+// per-tier, per-kind limits held in backend.config, read from the in-memory cache. It resolves
+// the caller's tier (guest vs durable) from the account, then counts its open+active games of that
+// kind. It reports false (not at the limit) when the limits config is not wired, the account is nil,
+// or the resolved cap is gamelimits.Unlimited. It backs the new-game gate (the handler aborts 409
+// game_limit_reached) and the lobby's at-limit flag.
+func (svc *Service) AtGameLimit(ctx context.Context, accountID uuid.UUID, kind gamelimits.Kind) (bool, error) {
+	if svc.limits == nil || accountID == uuid.Nil {
+		return false, nil
+	}
+	acc, err := svc.accounts.GetByID(ctx, accountID)
+	if err != nil {
+		return false, err
+	}
+	limit := svc.limits.LimitsFor(acc.IsGuest).Cap(kind)
+	if limit == gamelimits.Unlimited {
+		return false, nil
+	}
+	n, err := svc.store.CountActiveByKind(ctx, accountID, kind)
+	if err != nil {
+		return false, err
+	}
+	return n >= limit, nil
+}
+
 // walletContext resolves the payments gate inputs for an account on the current request: the
 // trusted execution context (from the session platform carried on ctx; absent ⇒ untrusted) and
 // the account's present identity sources (which chip/benefit segments are awake, §6).
@@ -338,6 +373,7 @@ func (svc *Service) Create(ctx context.Context, params CreateParams) (Game, erro
 		dropoutTiles:         params.DropoutTiles.String(),
 		multipleWordsPerTurn: params.MultipleWordsPerTurn,
 		vsAI:                 params.VsAI,
+		kind:                 params.Kind,
 	}
 	if err := svc.store.CreateGame(ctx, ins, seats, seeding.draws); err != nil {
 		return Game{}, err
@@ -401,6 +437,7 @@ func (svc *Service) OpenOrJoin(ctx context.Context, accountID uuid.UUID, params
 		multipleWordsPerTurn: params.MultipleWordsPerTurn,
 		status:               StatusOpen,
 		openDeadline:         &deadline,
+		kind:                 gamelimits.KindRandom,
 	}
 	// Decide the first move now by the official draw, with the not-yet-arrived opponent as a
 	// synthetic placeholder (uuid.Nil): the draw fixes who sits at seat 0 — and so moves
@@ -1386,14 +1423,6 @@ func (svc *Service) ListForLobby(ctx context.Context, accountID uuid.UUID) ([]Ga
 	return kept, nil
 }
 
-// CountActiveQuickGames reports how many in-progress quick games the account holds —
-// the count the simultaneous-game limit (MaxActiveQuickGames) is checked against. It
-// counts active and still-open quick games (including honest-AI ones) and excludes
-// friend games created by invitation and finished games. See Store.CountActiveQuickGames.
-func (svc *Service) CountActiveQuickGames(ctx context.Context, accountID uuid.UUID) (int, error) {
-	return svc.store.CountActiveQuickGames(ctx, accountID)
-}
-
 // HideGame hides a finished game from accountID's own lobby (it stays visible to the other
 // players); it is irreversible by design. Only a player of a finished game may hide it
 // (ErrNotAPlayer / ErrGameActive otherwise); hiding an already-hidden game is a no-op.