feat(ui): per-kind active-game limit lock on the New Game screen

Carry the caller's per-tier active-game caps and each game's kind on the
wire (Profile.game_limits + GameView.kind, additive FBS + gateway transcode
+ client codec, committed regen). The New Game screen counts the player's
active games per kind from the lobby and locks a capped start: an outline
button with a lock that opens a funnel modal instead of a game -- a sign-in
prompt for a guest, a "finish a current game first" notice for a signed-in
account (native Telegram popup, in-app modal elsewhere). The lock lifts via
the existing profile refetch after a guest->durable upgrade.

Remove the lobby's old at_game_limit New-Game tab disable + notice: the flag
(now the random-kind cap) conflicted with the per-kind lock -- it hid the
screen where the lock lives and wrongly blocked an unfulfilled kind. The New
Game tab is always enabled; the per-kind start lock is the only gate. The
at_game_limit wire field stays (unused by the client) for a later cleanup.

Tests: client lock logic + codec kind/game_limits roundtrip + gateway
transcode encode + native popup builders (unit); a mock e2e for the lock
badge and the modal.

gateway/internal/transcode/transcode_gamelimits_test.go

@@ -0,0 +1,61 @@
+package transcode_test
+
+import (
+	"context"
+	"net/http"
+	"testing"
+
+	"scrabble/gateway/internal/transcode"
+	fb "scrabble/pkg/fbs/scrabblefb"
+)
+
+// TestProfileGetEncodesGameLimits verifies the gateway forwards the backend's per-kind active-game
+// caps into the Profile payload — the caps the client's New-Game lock reads. The encode is not
+// exercised by the mock e2e (it bypasses the codec), so a dropped field would only surface here.
+func TestProfileGetEncodesGameLimits(t *testing.T) {
+	backend, cleanup := fakeBackend(t, func(w http.ResponseWriter, r *http.Request) {
+		if r.Method != http.MethodGet || r.URL.Path != "/api/v1/user/profile" {
+			t.Errorf("unexpected %s %q", r.Method, r.URL.Path)
+		}
+		_, _ = w.Write([]byte(`{"user_id":"u-1","display_name":"Kaya","preferred_language":"en",` +
+			`"game_limits":{"vs_ai":1,"random":1,"friends":0}}`))
+	})
+	defer cleanup()
+
+	reg := transcode.NewRegistry(backend, nil)
+	op, ok := reg.Lookup(transcode.MsgProfileGet)
+	if !ok {
+		t.Fatal("profile.get not registered")
+	}
+	payload, err := op.Handler(context.Background(), transcode.Request{UserID: "u-1"})
+	if err != nil {
+		t.Fatalf("handler: %v", err)
+	}
+
+	gl := fb.GetRootAsProfile(payload, 0).GameLimits(nil)
+	if gl == nil {
+		t.Fatal("profile carries no game_limits block")
+	}
+	if gl.VsAi() != 1 || gl.Random() != 1 || gl.Friends() != 0 {
+		t.Errorf("game limits = %d/%d/%d, want 1/1/0", gl.VsAi(), gl.Random(), gl.Friends())
+	}
+}
+
+// TestProfileGetNoGameLimits verifies a profile without a game_limits block encodes none (the
+// backend omits it only when the limits config is unwired; normally the block is present).
+func TestProfileGetNoGameLimits(t *testing.T) {
+	backend, cleanup := fakeBackend(t, func(w http.ResponseWriter, r *http.Request) {
+		_, _ = w.Write([]byte(`{"user_id":"u-1","display_name":"Kaya","preferred_language":"en"}`))
+	})
+	defer cleanup()
+
+	reg := transcode.NewRegistry(backend, nil)
+	op, _ := reg.Lookup(transcode.MsgProfileGet)
+	payload, err := op.Handler(context.Background(), transcode.Request{UserID: "u-1"})
+	if err != nil {
+		t.Fatalf("handler: %v", err)
+	}
+	if p := fb.GetRootAsProfile(payload, 0); p.GameLimits(nil) != nil {
+		t.Error("profile without a game_limits block unexpectedly carries one")
+	}
+}