fix(hint): stop the hint count going stale across games

The in-game hint badge re-fetched on entry and, for a game where it was the
player's turn, showed a too-high count that "reset" (e.g. back to 11) — the
wallet hint spent in another game was not reflected.

Root cause: the server sends one hints_remaining = per-game allowance + global
wallet, and the client cached that combined number per game. The wallet is
global, so spending a wallet hint in one game left every other game's cached
count stale (a my-turn game holds the stalest value: the opponent-moved delta
preserves the old number, whereas a game you just moved in re-cached a fresh
one). The backend allowance-then-wallet spend order was already correct.

Fix: split the two. StateView/HintResult gain a trailing wallet_balance field
(the global wallet alone); the client derives the per-game allowance as
hints_remaining - wallet_balance (stable, cacheable) and reads the wallet live
from the profile, refreshing it from every state/hint response. The badge is
allowance + live wallet, so a wallet hint anywhere updates every game at once.

- wire: scrabble.fbs StateView/HintResult + pkg/wire.BuildStateView (the single
  encoder for both the gateway transcode and the backend's event StateView),
  gateway encode + resp structs, regen.
- backend: game StateView/HintResult + service (GameState/Hint) + eventwire +
  notify PlayerState/encode + server DTOs.
- ui: lib/hints.ts (pure hintsLeft), Game.svelte (badge + syncWallet on
  load/hint, carry wallet_balance through applyMoveResult), codec/model, mock.
- docs: ARCHITECTURE §Hint.

Tests: hints.ts unit (incl. the staleness case), TestHintPolicy extended
(wallet_balance + allowance-first), gateway state/hint round-trips.

ui/src/game/Game.svelte

@@ -18,6 +18,7 @@
   import { centre, premiumGrid } from '../lib/premiums';
   import { variantNameKey } from '../lib/variants';
   import { alphabetLetters, hasAlphabet } from '../lib/alphabet';
+  import { hintsLeft } from '../lib/hints';
   import { shareOrDownloadGcg } from '../lib/share';
   import { getCachedGame, setCachedGame, setCachedDraft, type CachedGame } from '../lib/gamecache';
   import { patchLobbyGame } from '../lib/lobbycache';
@@ -132,6 +133,10 @@
   const playable = $derived(!!view && (view.game.status === 'active' || view.game.status === 'open'));
   const isMyTurn = $derived(!!view && playable && view.game.toMove === view.seat);
   const gameOver = $derived(!!view && view.game.status === 'finished');
+  // The hint badge: this game's allowance remaining plus the LIVE global wallet. Reading the
+  // wallet from the profile (not the per-game view snapshot) keeps it correct after a wallet
+  // hint was spent in another game (see lib/hints).
+  const hintCount = $derived(hintsLeft(view, app.profile?.hintBalance ?? 0));
   // RACK_SIZE mirrors the engine's rules.RackSize (7 for every current variant). The exchange
   // gate is only a UX guard: the backend stays the source of truth and rejects an under-supplied
   // exchange regardless (engine rejects when bag.Len() < rules.RackSize).
@@ -154,6 +159,13 @@
     return MOVE_LABELS.has(action) ? t(`move.${action}` as MessageKey) : action;
   }
 
+  // syncWallet adopts the server's authoritative hint-wallet balance into the global profile.
+  // The wallet is global, so keeping it live here (rather than per-game) is what stops the hint
+  // badge from going stale when a wallet hint was spent in another game.
+  function syncWallet(walletBalance: number) {
+    if (app.profile) app.profile.hintBalance = walletBalance;
+  }
+
   async function load() {
     try {
       // Ask for the alphabet table only on a per-variant cache miss (the first open of a
@@ -167,6 +179,7 @@
         gateway.draftGet(id).catch(() => ''),
       ]);
       view = st;
+      syncWallet(st.walletBalance);
       // Seed the unread flag from the authoritative state (the live stream only raises it).
       seedChatUnread(id, st.game.unreadChat);
       moves = hist.moves;
@@ -615,7 +628,16 @@
   // applyMoveResult renders the actor's own just-committed move from the response — the move, the
   // post-move game and the refilled rack — without a follow-up game.state + game.history.
   function applyMoveResult(r: MoveResult) {
-    view = { game: r.game, seat: r.move.player, rack: r.rack, bagLen: r.bagLen, hintsRemaining: view?.hintsRemaining ?? 0 };
+    view = {
+      game: r.game,
+      seat: r.move.player,
+      rack: r.rack,
+      bagLen: r.bagLen,
+      // A move is not a hint, so the per-game allowance and the wallet are unchanged: carry both
+      // forward (their difference is the stable allowance; the badge adds the live wallet).
+      hintsRemaining: view?.hintsRemaining ?? 0,
+      walletBalance: view?.walletBalance ?? 0,
+    };
     // The move result is an authoritative per-viewer view: a nudge the actor just answered by
     // moving is already cleared server-side, so reconcile the unread flag from it.
     seedChatUnread(id, r.game.unreadChat);
@@ -698,7 +720,8 @@
           recenter++;
         }
         if (isCoarse()) zoomed = true;
-        view = { ...view, hintsRemaining: h.hintsRemaining };
+        view = { ...view, hintsRemaining: h.hintsRemaining, walletBalance: h.walletBalance };
+        syncWallet(h.walletBalance);
         recompute();
       }
     } catch (e) {
@@ -1107,10 +1130,10 @@
   <TapConfirm
     triggerClass="tab"
     label={t('game.hint')}
-    disabled={busy || !isMyTurn || !connection.online || (view?.hintsRemaining ?? 0) <= 0}
+    disabled={busy || !isMyTurn || !connection.online || hintCount <= 0}
     onconfirm={doHint}
   >
-    <span class="sq">🛟{#if (view?.hintsRemaining ?? 0) > 0}<span class="badge">{view?.hintsRemaining}</span>{/if}</span>
+    <span class="sq">🛟{#if hintCount > 0}<span class="badge">{hintCount}</span>{/if}</span>
     <span class="lbl">{t('game.hint')}</span>
   </TapConfirm>
   {#if placement.pending.length > 0}