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.

backend/internal/server/banner.go

@@ -72,6 +72,7 @@ func (s *Server) profileResponse(ctx context.Context, acc account.Account) profi
 	}
 	r.Banner = s.bannerFor(ctx, acc, cxt, present)
 	r.Ads = s.adsFor(ctx, acc, cxt, present)
+	r.GameLimits = s.gameLimitsDTOFor(acc.IsGuest)
 	s.fillLinkedIdentities(ctx, &r, acc.ID)
 	r.DictVersions = s.currentDictVersions()
 	return r

backend/internal/server/dto.go

@@ -78,6 +78,18 @@ type profileResponse struct {
 	// Filled outside the pure projection (it reads the dictionary registry), so it is empty
 	// for callers that build the DTO without a Server. See Server.profileResponse.
 	DictVersions []dictVersion `json:"dict_versions,omitempty"`
+	// GameLimits carries the caller's tier active-game caps per kind (-1 = unlimited); the client
+	// counts its active games per kind from the lobby and locks a capped New-Game start. Filled
+	// outside the pure projection (it reads the limits config). See Server.profileResponse.
+	GameLimits *gameLimitsDTO `json:"game_limits,omitempty"`
+}
+
+// gameLimitsDTO is the caller's tier active-game caps per kind (-1 = unlimited), for the client's
+// per-kind New-Game lock. profileResponse.GameLimits carries it.
+type gameLimitsDTO struct {
+	VsAI    int `json:"vs_ai"`
+	Random  int `json:"random"`
+	Friends int `json:"friends"`
 }
 
 // dictVersion pairs a game variant's stable label (engine.Variant.String) with its current
@@ -134,7 +146,11 @@ type gameDTO struct {
 	MultipleWordsPerTurn bool   `json:"multiple_words_per_turn"`
 	// VsAI marks an honest-AI game: the opponent is shown as 🤖 and chat/nudge/add-friend
 	// are suppressed in the client.
-	VsAI      bool   `json:"vs_ai"`
+	VsAI bool `json:"vs_ai"`
+	// Kind is the game's origin for the active-game limits (game.Kind): 0 unknown (a pre-existing
+	// game), 1 vs_ai, 2 random, 3 friends. The lobby counts active games per kind to lock a capped
+	// New-Game start.
+	Kind      int    `json:"kind"`
 	MoveCount int    `json:"move_count"`
 	EndReason string `json:"end_reason"`
 	// LastActivityUnix is the lobby sort key: the current turn's start for an active
@@ -277,6 +293,7 @@ func gameDTOFromGame(g game.Game) gameDTO {
 		TurnTimeoutSecs:      int(g.TurnTimeout.Seconds()),
 		MultipleWordsPerTurn: g.MultipleWordsPerTurn,
 		VsAI:                 g.VsAI,
+		Kind:                 int(g.Kind),
 		MoveCount:            g.MoveCount,
 		EndReason:            g.EndReason,
 		LastActivityUnix:     last.Unix(),

backend/internal/server/handlers_admin_limits.go

@@ -9,6 +9,16 @@ import (
 	"scrabble/backend/internal/gamelimits"
 )
 
+// gameLimitsDTOFor resolves the caller's tier active-game caps into the profile DTO, so the client
+// can lock a capped New-Game start per kind. It returns nil when the limits config is not wired.
+func (s *Server) gameLimitsDTOFor(isGuest bool) *gameLimitsDTO {
+	if s.gamelimits == nil {
+		return nil
+	}
+	l := s.gamelimits.LimitsFor(isGuest)
+	return &gameLimitsDTO{VsAI: l.VsAI, Random: l.Random, Friends: l.Friends}
+}
+
 // 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) {