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.

ui/src/lib/model.ts

@@ -62,6 +62,11 @@ export interface GameView {
    *  badge can be coloured apart from the nudge-only case. Same REST-seeded lifecycle as
    *  unreadChat; the live-event GameView leaves it false. */
   unreadMessages: boolean;
+  /** The game's origin for the active-game limits: 0 unknown (a pre-existing game, never gated),
+   *  1 vs_ai, 2 random, 3 friends. The lobby counts active games per kind against the caller's
+   *  per-kind limits (Profile.gameLimits) to lock a capped New-Game start. Local/offline games
+   *  leave it 0 (they never count toward the online limits). */
+  kind: number;
   seats: Seat[];
 }
 
@@ -234,6 +239,21 @@ export interface Profile {
    *  offline-capable PWA reads it to preload the matching dawg for each enabled variant off
    *  this cold-start response. Empty only in a degenerate deployment with no resident dictionary. */
   dictVersions: Partial<Record<Variant, string>>;
+  /** The caller's tier active-game caps per kind (-1 = unlimited); the New-Game screen counts the
+   *  player's active games per kind from the lobby and locks a capped start. Absent from an old
+   *  backend (then no lock applies). */
+  gameLimits?: GameLimits;
+}
+
+/** GameKind labels the game_kind wire values for the active-game limits. */
+export const GameKind = { vsAi: 1, random: 2, friends: 3 } as const;
+
+/** The caller's tier active-game caps per kind (-1 = unlimited), resolved to the guest or durable
+ *  tier server-side. The New-Game screen locks a start whose kind is at its cap. */
+export interface GameLimits {
+  vsAi: number;
+  random: number;
+  friends: number;
 }
 
 /** The post-move interstitial-ad config for the client-mirrored gate: the cooldowns (seconds) and