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/game/ChatScreen.svelte

@@ -0,0 +1,93 @@
+<script lang="ts">
+  import { onMount } from 'svelte';
+  import Screen from '../components/Screen.svelte';
+  import Chat from './Chat.svelte';
+  import { gateway } from '../lib/gateway';
+  import { app, handleError, clearChatUnread } from '../lib/app.svelte';
+  import { t } from '../lib/i18n/index.svelte';
+  import type { ChatMessage, StateView } from '../lib/model';
+
+  // The chat is its own screen (Stage 17), so the soft keyboard simply resizes the viewport with
+  // the input pinned to the bottom — no modal relayout jank. It loads the game state (for the
+  // turn-based chat/nudge toggle) and the message list, and clears the unread badge while open.
+  let { id }: { id: string } = $props();
+
+  let view = $state<StateView | null>(null);
+  let messages = $state<ChatMessage[]>([]);
+  let busy = $state(false);
+  let tick = $state(0);
+
+  const myId = $derived(app.session?.userId ?? '');
+  const isMyTurn = $derived(!!view && view.game.status === 'active' && view.game.toMove === view.seat);
+  const nudgeCooldownSecs = 3600;
+  // The nudge is one-per-hour-per-game and clears once the player chats (engagement); the
+  // backend stays authoritative, so a move-based reset is left to it.
+  const nudgeOnCooldown = $derived.by(() => {
+    void tick;
+    let lastNudge = 0;
+    let lastChat = 0;
+    for (const m of messages) {
+      if (m.senderId !== myId) continue;
+      if (m.kind === 'nudge') lastNudge = Math.max(lastNudge, m.createdAtUnix);
+      else lastChat = Math.max(lastChat, m.createdAtUnix);
+    }
+    if (lastNudge === 0 || Date.now() / 1000 - lastNudge >= nudgeCooldownSecs) return false;
+    return lastChat <= lastNudge;
+  });
+
+  async function refresh() {
+    try {
+      messages = await gateway.chatList(id);
+      clearChatUnread(id);
+    } catch {
+      /* best-effort */
+    }
+  }
+  onMount(async () => {
+    try {
+      view = await gateway.gameState(id, false);
+    } catch (e) {
+      handleError(e);
+    }
+    await refresh();
+  });
+
+  // Live: refresh (and keep the unread cleared) on a chat / nudge for this game.
+  $effect(() => {
+    const e = app.lastEvent;
+    if (!e) return;
+    if ((e.kind === 'chat_message' && e.message.gameId === id) || (e.kind === 'nudge' && e.gameId === id)) {
+      void refresh();
+    }
+  });
+  // Re-evaluate the nudge cooldown on a timer so the control re-enables on time.
+  $effect(() => {
+    const h = setInterval(() => (tick += 1), 20000);
+    return () => clearInterval(h);
+  });
+
+  async function sendChat(text: string) {
+    busy = true;
+    try {
+      messages = [...messages, await gateway.chatPost(id, text)];
+    } catch (e) {
+      handleError(e);
+    } finally {
+      busy = false;
+    }
+  }
+  async function nudge() {
+    busy = true;
+    try {
+      messages = [...messages, await gateway.nudge(id)];
+    } catch (e) {
+      handleError(e);
+    } finally {
+      busy = false;
+    }
+  }
+</script>
+
+<Screen title={t('game.chat')} back={`/game/${id}`} scroll={false} column>
+  <Chat {messages} {myId} {busy} myTurn={isMyTurn} {nudgeOnCooldown} onsend={sendChat} onnudge={nudge} />
+</Screen>