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).

PLAN.md

@@ -1403,10 +1403,16 @@ provided cert) at the contour caddy; prod VPN; rollback.
     - **Edge-swipe back** (`Screen.svelte`): a left-edge rightward drag navigates to `back`
       (touch/pen only, armed only from ≤24px so it never fights the board's gestures; skipped
       inside Telegram, which has its own back).
-    - **Chat soft-keyboard** is a **bottom-sheet** `Modal` lifted above the keyboard by a
-      `transform` driven by `visualViewport` (compositor-only — the board behind and the sheet
-      no longer relayout as the keyboard animates). iOS-specific; needs on-device fine-tuning.
-      The native `Keyboard.setResizeMode('none')` path waits for Capacitor (not yet wired).
+    - **Chat + word-check are now their own routed screens** (`/game/:id/chat`,
+      `/game/:id/check`, header back to the game, no tab-bar) so the soft keyboard simply resizes
+      the **visible viewport** — mirrored into a `--vvh` CSS var the `Screen` height uses, since
+      iOS doesn't shrink `dvh` for the keyboard — with the input pinned to the bottom: no modal
+      relayout, no page jump (this superseded a first bottom-sheet-`Modal` attempt). New chat
+      messages raise an **unread badge** on the in-game hamburger + the Chat menu row (per game,
+      cleared on open), mirroring the lobby badge; the chat screen is routable for a future
+      Telegram deep-link. The TG-fullscreen header was also finalised over a couple of review
+      passes: title + menu as a centred pair inside Telegram's nav band (between `--tg-safe-top`
+      and `--tg-content-top`), with a small padding bump so the native controls aren't flush.
     - **Tests backfilled** for the merged round-6 work: e2e for the in-game "✓ in friends" item
       and a board→board tile relocation; codec units for `last_activity_unix` + `OutgoingRequestList`.
     - **Deferred to the next PR (agreed):** #4 enrich the out-of-app "your turn" / game-end push