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/screens/NewGame.svelte

@@ -2,7 +2,11 @@
   import { onMount } from 'svelte';
   import Screen from '../components/Screen.svelte';
   import Modal from '../components/Modal.svelte';
+  import GameLimitModal from '../components/GameLimitModal.svelte';
   import PinPad, { type PinResult } from '../components/PinPad.svelte';
+  import { getLobby } from '../lib/lobbycache';
+  import { isKindLocked } from '../lib/gamelimits';
+  import { GameKind } from '../lib/model';
   import { gateway } from '../lib/gateway';
   import { app, handleError, showToast } from '../lib/app.svelte';
   import { connection } from '../lib/connection.svelte';
@@ -51,6 +55,14 @@
   // or a random human via auto-match. AI is the default.
   let opponent = $state<'ai' | 'random'>('ai');
 
+  // Active-game limit lock: the auto-start kind (vs_ai or random by the opponent choice) tested
+  // against the caller's per-kind cap over the online lobby's active games. A capped start opens the
+  // funnel modal instead of enqueuing (the server also refuses it); offline never locks (local
+  // games are uncapped). The lock lifts when the profile is re-fetched after a guest→durable upgrade.
+  const autoKind = $derived(opponent === 'ai' ? GameKind.vsAi : GameKind.random);
+  const autoLocked = $derived(!offlineMode.active && isKindLocked(getLobby(false)?.games ?? [], app.profile?.gameLimits, autoKind));
+  let limitOpen = $state(false);
+
   // --- auto-match ---
   // Enqueue drops the player straight into a real game — a freshly opened one awaiting an
   // opponent, or another player's open game they just joined — so we navigate into it at once
@@ -329,9 +341,16 @@
         {#if opponent === 'random'}<p class="searchhint">{t('new.searchHint')}</p>{/if}
         <button
           class="invite"
-          disabled={!selectedAuto || (!connection.online && !offlineMode.active) || starting}
-          onclick={() => selectedAuto && find(selectedAuto)}
-        >{t('new.start')}</button>
+          class:locked={autoLocked}
+          disabled={(autoLocked ? false : !selectedAuto) || (!connection.online && !offlineMode.active) || starting}
+          onclick={() => (autoLocked ? (limitOpen = true) : selectedAuto && find(selectedAuto))}
+        >{#if autoLocked}🔒 {/if}{t('new.start')}</button>
+        <GameLimitModal
+          open={limitOpen}
+          {guest}
+          onclose={() => (limitOpen = false)}
+          onlogin={() => { limitOpen = false; navigate('/settings'); }}
+        />
       {:else if offlineMode.active}
         <!-- Offline pass-and-play (hotseat): a device-local 2-4 human game. The host sets a master
              PIN first (it gates the roster); each seat may add its own PIN. -->
@@ -660,6 +679,12 @@
   .invite:disabled {
     opacity: 0.5;
   }
+  /* A capped start (the active-game limit): an outline instead of the filled CTA, with a 🔒, so it
+     reads as gated rather than ready. Tapping it opens the funnel modal, never a game. */
+  .invite.locked {
+    background: transparent;
+    color: var(--accent);
+  }
   .searchhint {
     margin: 0;
     text-align: center;