Chat + word-check as their own screens; in-game unread badge (review item 7)

- Chat and word-check are now routed screens (/game/:id/chat, /game/:id/check) with a
  header back to the game and no tab-bar, replacing their modals. The soft keyboard just
  resizes the visible viewport (tracked into --vvh, which the Screen height uses since iOS
  does not shrink dvh for the keyboard) with the input pinned to the bottom: no modal
  relayout, no page jump. Supersedes the earlier bottom-sheet Modal attempt.
- A new chat message raises an unread badge on the in-game hamburger + the Chat menu row
  (per game, cleared on opening the chat), mirroring the lobby badge.
- TG native back + the header back chevron return chat/check to their game.
- Exposes --tg-safe-top (device notch) for the finalised TG-fullscreen header.

Tests: e2e for chat/check opening as their own screens + back. Docs: PLAN, FUNCTIONAL(+ru).

ui/src/App.svelte

@@ -15,6 +15,8 @@
   import Friends from './screens/Friends.svelte';
   import Stats from './screens/Stats.svelte';
   import Game from './game/Game.svelte';
+  import ChatScreen from './game/ChatScreen.svelte';
+  import CheckScreen from './game/CheckScreen.svelte';
 
   onMount(() => {
     void bootstrap();
@@ -25,8 +27,11 @@
   // back chevron is hidden in Telegram (Header.svelte) so only the native one shows.
   $effect(() => {
     if (!insideTelegram()) return;
-    const name = router.route.name;
-    telegramBackButton(name !== 'lobby' && name !== 'login', () => navigate('/'));
+    const r = router.route;
+    // The chat / check sub-screens step back to their game; every other sub-screen to the lobby.
+    const sub = r.name === 'gameChat' || r.name === 'gameCheck';
+    const target = sub ? `/game/${r.params.id}` : '/';
+    telegramBackButton(r.name !== 'lobby' && r.name !== 'login', () => navigate(target));
   });
 
   // Screen transitions: the lobby is the navigation root. Entering a screen from the
@@ -63,6 +68,10 @@
           <NewGame />
         {:else if router.route.name === 'game'}
           <Game id={router.route.params.id} />
+        {:else if router.route.name === 'gameChat'}
+          <ChatScreen id={router.route.params.id} />
+        {:else if router.route.name === 'gameCheck'}
+          <CheckScreen id={router.route.params.id} />
         {:else if router.route.name === 'profile'}
           <Profile />
         {:else if router.route.name === 'settings'}