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.

backend/internal/inttest/game_test.go

@@ -408,6 +408,15 @@ func TestHintPolicy(t *testing.T) {
 	if _, err := svc.Hint(ctx, g.ID, seats[0]); err != nil { // spends the allowance
 		t.Fatalf("first hint: %v", err)
 	}
+	// The allowance is spent before the wallet: with an empty wallet, the state now reports no
+	// hints left and a zero wallet, so the per-game allowance (HintsRemaining-WalletBalance) is 0.
+	st, err := svc.GameState(ctx, g.ID, seats[0])
+	if err != nil {
+		t.Fatalf("state: %v", err)
+	}
+	if st.HintsRemaining != 0 || st.WalletBalance != 0 {
+		t.Errorf("after allowance hint: hints=%d wallet=%d, want 0/0", st.HintsRemaining, st.WalletBalance)
+	}
 	if _, err := svc.Hint(ctx, g.ID, seats[0]); !errors.Is(err, game.ErrNoHintsLeft) {
 		t.Fatalf("second hint = %v, want ErrNoHintsLeft", err)
 	}
@@ -416,8 +425,9 @@ func TestHintPolicy(t *testing.T) {
 	if err != nil {
 		t.Fatalf("wallet hint: %v", err)
 	}
-	if res.HintsRemaining != 1 {
-		t.Errorf("hints remaining = %d, want 1", res.HintsRemaining)
+	// The allowance stays exhausted; the wallet dropped 2->1, and WalletBalance carries it alone.
+	if res.HintsRemaining != 1 || res.WalletBalance != 1 {
+		t.Errorf("wallet hint: hints=%d wallet=%d, want 1/1", res.HintsRemaining, res.WalletBalance)
 	}
 
 	off, err := svc.Create(ctx, game.CreateParams{