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.

docs/ARCHITECTURE.md

@@ -347,7 +347,13 @@ Key points:
   placement and leaves the commit to the player. When the rack has no legal move the
   service spends **nothing** and returns `ErrNoHintAvailable` — surfaced as the distinct
   result code `no_hint_available` (separate from `hint_unavailable`) so the UI can say
-  "no options" rather than "no hints left".
+  "no options" rather than "no hints left". The hint count shown to the player is the
+  per-game allowance remaining **plus** the global wallet; because the wallet is global,
+  `game.state`/`game.hint` carry it as a separate `wallet_balance` field beside the combined
+  `hints_remaining`, so the client derives the per-game allowance (`hints_remaining -
+  wallet_balance`, which it may cache per game) and reads the wallet **live** from the
+  profile — otherwise a wallet hint spent in one game would leave a stale, too-high count
+  cached on every other game.
 - **Word-check tool**: unlimited dictionary lookups against the game's pinned
   dictionary; each result offers a **complaint** (complainant, game, variant,
   dict_version, word, the disputed result, an optional note) that lands in the admin