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/gamelimits/gamelimits.go

@@ -0,0 +1,149 @@
+// Package gamelimits holds the per-tier, per-kind active-game caps (a guest funnel) with a hot
+// in-memory cache over the single-row backend.config, so a login or a game-create never queries it.
+package gamelimits
+
+import (
+	"context"
+	"database/sql"
+	"fmt"
+	"sync"
+
+	"github.com/go-jet/jet/v2/postgres"
+
+	"scrabble/backend/internal/postgres/jet/backend/model"
+	"scrabble/backend/internal/postgres/jet/backend/table"
+)
+
+// Kind is a game's kind, mirroring backend.games.game_kind. 0 (unknown) is an untagged game, never gated.
+type Kind int16
+
+const (
+	KindUnknown Kind = 0
+	KindVsAI    Kind = 1
+	KindRandom  Kind = 2
+	KindFriends Kind = 3
+)
+
+// String returns the kind's label for admin game lists and logs.
+func (k Kind) String() string {
+	switch k {
+	case KindVsAI:
+		return "vs_ai"
+	case KindRandom:
+		return "random"
+	case KindFriends:
+		return "friends"
+	default:
+		return "unknown"
+	}
+}
+
+// Unlimited is the limit sentinel meaning no cap.
+const Unlimited = -1
+
+// Limits is a tier's active-game caps per kind (-1 = unlimited).
+type Limits struct {
+	VsAI    int
+	Random  int
+	Friends int
+}
+
+// Cap returns the limit for kind; the unknown kind is never capped.
+func (l Limits) Cap(kind Kind) int {
+	switch kind {
+	case KindVsAI:
+		return l.VsAI
+	case KindRandom:
+		return l.Random
+	case KindFriends:
+		return l.Friends
+	default:
+		return Unlimited
+	}
+}
+
+// Config is the per-tier limit config (the single backend.config row).
+type Config struct {
+	Guest   Limits
+	Durable Limits
+}
+
+// Store reads and writes the single-row backend.config.
+type Store struct{ db *sql.DB }
+
+// NewStore constructs a Store over db.
+func NewStore(db *sql.DB) *Store { return &Store{db: db} }
+
+func (s *Store) load(ctx context.Context) (Config, error) {
+	var c model.Config
+	if err := postgres.SELECT(table.Config.AllColumns).FROM(table.Config).LIMIT(1).
+		QueryContext(ctx, s.db, &c); err != nil {
+		return Config{}, fmt.Errorf("gamelimits: load config: %w", err)
+	}
+	return Config{
+		Guest:   Limits{VsAI: int(c.GuestVsAiLimit), Random: int(c.GuestRandomLimit), Friends: int(c.GuestFriendsLimit)},
+		Durable: Limits{VsAI: int(c.DurableVsAiLimit), Random: int(c.DurableRandomLimit), Friends: int(c.DurableFriendsLimit)},
+	}, nil
+}
+
+func (s *Store) save(ctx context.Context, c Config) error {
+	if _, err := s.db.ExecContext(ctx,
+		`UPDATE backend.config SET guest_vs_ai_limit=$1, guest_random_limit=$2, guest_friends_limit=$3,
+		   durable_vs_ai_limit=$4, durable_random_limit=$5, durable_friends_limit=$6 WHERE only_row`,
+		c.Guest.VsAI, c.Guest.Random, c.Guest.Friends, c.Durable.VsAI, c.Durable.Random, c.Durable.Friends); err != nil {
+		return fmt.Errorf("gamelimits: save config: %w", err)
+	}
+	return nil
+}
+
+// Service fronts the config with an in-memory cache (single-instance, matching the deploy). Load
+// once at startup; Update after an admin edit refreshes it in place.
+type Service struct {
+	store *Store
+	mu    sync.RWMutex
+	cfg   Config
+}
+
+// NewService constructs a Service over store. Call Load before serving.
+func NewService(store *Store) *Service { return &Service{store: store} }
+
+// Load reads the config into the cache. Call once at startup.
+func (s *Service) Load(ctx context.Context) error {
+	c, err := s.store.load(ctx)
+	if err != nil {
+		return err
+	}
+	s.set(c)
+	return nil
+}
+
+func (s *Service) set(c Config) {
+	s.mu.Lock()
+	s.cfg = c
+	s.mu.Unlock()
+}
+
+// Get returns the cached config.
+func (s *Service) Get() Config {
+	s.mu.RLock()
+	defer s.mu.RUnlock()
+	return s.cfg
+}
+
+// LimitsFor returns the cached limits for the account tier (guest or durable).
+func (s *Service) LimitsFor(isGuest bool) Limits {
+	c := s.Get()
+	if isGuest {
+		return c.Guest
+	}
+	return c.Durable
+}
+
+// Update saves the config and refreshes the cache in place (the admin edit).
+func (s *Service) Update(ctx context.Context, c Config) error {
+	if err := s.store.save(ctx, c); err != nil {
+		return err
+	}
+	s.set(c)
+	return nil
+}

backend/internal/gamelimits/gamelimits_test.go

@@ -0,0 +1,44 @@
+package gamelimits
+
+import "testing"
+
+// TestLimitsCap checks Cap maps each kind to its field and leaves the unknown kind uncapped, so a
+// untagged game (game_kind 0) is never gated.
+func TestLimitsCap(t *testing.T) {
+	l := Limits{VsAI: 1, Random: 2, Friends: 3}
+	for _, tc := range []struct {
+		kind Kind
+		want int
+	}{
+		{KindVsAI, 1},
+		{KindRandom, 2},
+		{KindFriends, 3},
+		{KindUnknown, Unlimited},
+		{Kind(99), Unlimited},
+	} {
+		if got := l.Cap(tc.kind); got != tc.want {
+			t.Errorf("Cap(%d) = %d, want %d", tc.kind, got, tc.want)
+		}
+	}
+}
+
+// TestServiceLimitsForTier checks LimitsFor selects the guest or durable tier from the cached config.
+func TestServiceLimitsForTier(t *testing.T) {
+	svc := NewService(nil)
+	svc.set(Config{
+		Guest:   Limits{VsAI: 1, Random: 1, Friends: 0},
+		Durable: Limits{VsAI: 10, Random: 10, Friends: 10},
+	})
+
+	if got := svc.LimitsFor(true); got != (Limits{VsAI: 1, Random: 1, Friends: 0}) {
+		t.Errorf("guest limits = %+v, want {1 1 0}", got)
+	}
+	if got := svc.LimitsFor(false); got != (Limits{VsAI: 10, Random: 10, Friends: 10}) {
+		t.Errorf("durable limits = %+v, want {10 10 10}", got)
+	}
+	// The unlimited sentinel resolves through the tier too.
+	svc.set(Config{Durable: Limits{VsAI: Unlimited, Random: Unlimited, Friends: Unlimited}})
+	if got := svc.LimitsFor(false).Cap(KindVsAI); got != Unlimited {
+		t.Errorf("durable vs_ai cap = %d, want unlimited (-1)", got)
+	}
+}