docs: sync VK theme-follow + home-bar safe-area into FUNCTIONAL/UI_DESIGN/.claude

The #142 polish shipped three VK behaviours the docs never picked up (they were
written for the #140 auth MVP):

- the "auto" theme follows the VK client light/dark (VKWebAppUpdateConfig), not
  the VK webview's prefers-color-scheme;
- the layout clears the VK mobile home bar via CSS env() max'd with the VK bridge
  insets (VKWebAppUpdateInsets) — the bridge value is needed on Android, where the
  VK webview exposes no env() inset (.claude said env() handled it alone);
- share/copy route through the bridge (VKWebAppShare/VKWebAppCopyText).

FUNCTIONAL (+_ru) VK paragraph gains theme+home-bar parity with the Telegram one;
UI_DESIGN gets a VK-integration note beside the Telegram one.

docs/UI_DESIGN.md

@@ -113,6 +113,13 @@ dismisses as soon as the lobby is ready. The pure layout and timing live in `lib
   by a background suspend reconnects silently on return — the connection banner is
   suppressed while hidden and for a short grace after resume (visibilitychange +
   pageshow/pagehide + Telegram `activated`/`deactivated`).
+- **VK integration** (`lib/vk.ts`): inside the VK Mini App the **auto** theme follows the VK
+  client's light/dark (`VKWebAppUpdateConfig`), as the VK webview's `prefers-color-scheme` does not
+  track it; explicit light/dark prefs still win. The device safe-area comes from CSS
+  `env(safe-area-inset-*)` (the meta already sets `viewport-fit=cover`) **max'd** with the VK bridge
+  insets (`VKWebAppUpdateInsets`, needed on Android, where the VK webview exposes no `env()` inset),
+  so the layout clears the VK home bar. Share and copy route through the bridge (`VKWebAppShare` /
+  `VKWebAppCopyText`) because `navigator.share` / `clipboard` are absent in the desktop VK iframe.
 
 ## Tiles & board