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/mock/client.ts

@@ -33,6 +33,7 @@ import type {
   MoveResult,
   OutgoingList,
   Profile,
+  GameLimits,
   ProfileUpdate,
   PushEvent,
   Session,
@@ -105,8 +106,6 @@ export class MockGateway implements GatewayClient {
   private feedbackReplyUnread = false;
   // The most recently opened auto-match game still awaiting an opponent, for the e2e join hook.
   private openGameId: string | null = null;
-  // gameLimit forces the lobby's at_game_limit flag for the e2e (window.__mock.setGameLimit).
-  private gameLimit = false;
   private friends: AccountRef[] = MOCK_FRIENDS.map((f) => ({ ...f }));
   private incoming: AccountRef[] = MOCK_INCOMING.map((f) => ({ ...f }));
   private outgoing: AccountRef[] = [];
@@ -267,7 +266,7 @@ export class MockGateway implements GatewayClient {
   async gamesList(): Promise<GameList> {
     return {
       games: [...this.games.values()].map((g) => structuredClone(g.view)),
-      atGameLimit: this.gameLimit,
+      atGameLimit: false,
     };
   }
 
@@ -293,6 +292,7 @@ export class MockGateway implements GatewayClient {
           vsAi: true,
           unreadChat: false,
           unreadMessages: false,
+          kind: 1,
           seats: [
             { seat: 0, accountId: ME, displayName: 'You', score: 0, hintsUsed: 0, isWinner: false },
             { seat: 1, accountId: 'robot', displayName: 'Robot', score: 0, hintsUsed: 0, isWinner: false },
@@ -326,6 +326,7 @@ export class MockGateway implements GatewayClient {
         vsAi: false,
         unreadChat: false,
         unreadMessages: false,
+        kind: 2,
         seats: [
           { seat: 0, accountId: ME, displayName: 'You', score: 0, hintsUsed: 0, isWinner: false },
           { seat: 1, accountId: '', displayName: '', score: 0, hintsUsed: 0, isWinner: false },
@@ -377,13 +378,13 @@ export class MockGateway implements GatewayClient {
     if (this.openGameId) this.seatOpponent(this.openGameId);
   }
 
-  // setGameLimit forces the lobby's at_game_limit flag (the e2e hook
-  // window.__mock.setGameLimit), then nudges the lobby to re-fetch games.list so the
-  // "New Game" button and the notice update in place. The lobby refetches on any
-  // non-heartbeat event; an unrecognised notify sub triggers only that refetch.
-  setGameLimit(v: boolean): void {
-    this.gameLimit = v;
-    this.emit({ kind: 'notify', sub: 'game_limit' });
+  // setGameLimits overrides the profile's per-kind active-game caps (the e2e hook
+  // window.__mock.setGameLimits), then emits a profile notify so the app re-fetches the profile and
+  // the New-Game screen's per-kind lock recomputes in place — the same path a guest→durable upgrade
+  // takes to lift the lock.
+  setGameLimits(l: GameLimits): void {
+    this.profile.gameLimits = { ...l };
+    this.emit({ kind: 'notify', sub: 'profile' });
   }
 
   async lobbyPoll(): Promise<MatchResult> {