fix(ui): keep lobby/game caches fresh across screens (no stale-status flash)

The per-screen in-memory caches (lobbycache, gamecache) were refreshed only by
the screen that owns them while it was mounted, so a state change that crossed
screens left the other screen's cache stale and it visibly redrew on the next
navigation:

- game -> lobby: the player's own move advanced the game cache but not the lobby
  snapshot, and an own move carries no self-directed push event, so returning to
  the lobby painted the pre-move status until the background refresh corrected it.
- lobby -> game (and from any other screen): an opponent's move / game-over
  refreshed the lobby (while mounted) but never the per-game cache, so opening
  that game flashed the pre-move board.

Make cache freshness cross-screen, owned by the single global stream handler
that runs for every live event regardless of the mounted screen:

- patchLobbyGame upserts the affected game's GameView into the lobby snapshot;
  the global handler calls it on opponent_moved / game_over / opponent_joined and
  on a match_found / game_started seed (so a game started elsewhere is present
  too). The game board still mirrors the player's own move and its own load() —
  the two updates no live event carries.
- advanceCached (a pure wrapper over the existing delta reducers) advances a
  not-currently-viewed game's cache from opponent_moved / game_over; the game in
  view is skipped so its mounted board stays the sole owner (no double apply).

End-state behaviour is unchanged (the background refresh always reconciled);
this removes the transient stale frame. Unit-tested patchLobbyGame and
advanceCached; docs/UI_DESIGN.md updated.

ui/src/game/Game.svelte

@@ -20,6 +20,7 @@
   import { alphabetLetters, hasAlphabet } from '../lib/alphabet';
   import { shareOrDownloadGcg } from '../lib/share';
   import { getCachedGame, setCachedGame, type CachedGame } from '../lib/gamecache';
+  import { patchLobbyGame } from '../lib/lobbycache';
   import { applyGameOver, applyMoveDelta, type DeltaResult } from '../lib/gamedelta';
   import { telegramClosingConfirmation, telegramHaptic } from '../lib/telegram';
   import {
@@ -138,6 +139,9 @@
       view = st;
       moves = hist.moves;
       setCachedGame(id, st, hist.moves);
+      // Mirror the fresh status into the lobby snapshot so returning there shows it without a
+      // stale flash before the lobby's own background refresh lands.
+      patchLobbyGame(st.game);
       selected = null;
       await applyDraft(st);
       recompute();
@@ -559,6 +563,7 @@
     view = { game: r.game, seat: r.move.player, rack: r.rack, bagLen: r.bagLen, hintsRemaining: view?.hintsRemaining ?? 0 };
     moves = [...moves, r.move];
     setCachedGame(id, view, moves);
+    patchLobbyGame(r.game);
     rackIds = r.rack.map((_, i) => i);
     placement = newPlacement(r.rack);
     selected = null;