feat(hint): unify the vs_ai idle hint online + offline (server-enforced, monotonic)

Online vs_ai hints were broken: #207 made the vs_ai hint button always-enabled and
wallet-free (assuming all vs_ai = the offline idle-gate), but the backend still served
online vs_ai from the allowance/wallet, so over-clicking hit ErrNoHintsLeft -> a generic
error toast. Now online vs_ai uses the SAME idle-gate model as offline.

Backend: Hint() for a vs_ai game skips the allowance/wallet and increments no hints_used
(owner: vs_ai counts toward no hint statistic), and is idle-gated from the SERVER clock --
it returns ErrHintLocked (code hint_locked) until the robot's last move + 30 min, else
serves the top move. GameState/StateView expose hint_unlock_left_seconds (server-computed
seconds remaining; 0 for a human game / first move / not-your-turn). Pure helper
hintUnlockLeftSeconds unit-tested.

Wire: StateView gains hint_unlock_left_seconds (FlatBuffers, additive); pkg/wire + gateway
transcode carry it (round-trip test).

Client: the gate counts down from a MONOTONIC clock (performance.now()) anchored to the
source's seconds-left when it lands (on load from the view; to the full window when the
robot moves), so a client clock change cannot skew it and a relaunch re-reads a fresh
value. The vs_ai hint button (online + offline) shows the lock + toast; doHint handles the
hint_locked backstop by re-syncing. Replaces #207's absolute hintUnlockAtMs on the
wire/model/delta (the offline record keeps the absolute for persistence; the view exposes
seconds-left).

Docs FUNCTIONAL(+_ru)/ARCHITECTURE updated. Verified: go build/vet + game/server/transcode
tests (+ new hintUnlockLeftSeconds); ui check 0 / unit 490 / e2e 198 (one pre-existing
webkit offline flake) / app entry 114.2/115.

backend/internal/game/service.go

@@ -1058,10 +1058,31 @@ func (svc *Service) MarkChangesApplied(ctx context.Context, variant engine.Varia
 	return svc.store.MarkChangesApplied(ctx, variant.String(), version)
 }
 
-// Hint reveals the top-scoring legal play for the requesting player on their
-// turn, spending one hint from their per-game allowance and then their profile
-// wallet. It returns ErrHintsDisabled, ErrNoHintsLeft or ErrNoHintAvailable as
-// appropriate.
+// hintIdleWindow is how long a vs_ai player must be stuck on a turn (since the robot's last move)
+// before the idle hint unlocks. Mirrors the offline client (lib/hints HINT_GATE_MS = 30 min).
+const hintIdleWindow = 30 * time.Minute
+
+// hintUnlockLeftSeconds is the seconds until g's vs_ai idle hint unlocks for seat, measured from now:
+// the robot's last move (the current turn's start, on the human's turn) plus the window, floored at
+// 0 and ceiled to whole seconds. It is 0 for a non-vs_ai game, when it is not seat's turn, or on the
+// human's first move (MoveCount 0, no robot move yet) — the gate is an anti-frustration aid, not a
+// first-move tax. The client anchors a monotonic countdown to it, so a client clock cannot skew it.
+func hintUnlockLeftSeconds(g Game, seat int, now time.Time) int {
+	if !g.VsAI || g.ToMove != seat || g.MoveCount < 1 {
+		return 0
+	}
+	left := g.TurnStartedAt.Add(hintIdleWindow).Sub(now)
+	if left <= 0 {
+		return 0
+	}
+	return int((left + time.Second - 1) / time.Second) // ceil to whole seconds (no math import)
+}
+
+// Hint reveals the top-scoring legal play for the requesting player on their turn. For a human game
+// it spends one hint from the per-game allowance then the profile wallet (ErrHintsDisabled /
+// ErrNoHintsLeft / ErrNoHintAvailable). For a vs_ai game the hint is unlimited and wallet-free but
+// idle-gated from the server clock (ErrHintLocked until the window elapses) and counts toward no
+// hint statistic.
 func (svc *Service) Hint(ctx context.Context, gameID, accountID uuid.UUID) (HintResult, error) {
 	pre, err := svc.store.GetGame(ctx, gameID)
 	if err != nil {
@@ -1080,6 +1101,26 @@ func (svc *Service) Hint(ctx context.Context, gameID, accountID uuid.UUID) (Hint
 	if !pre.HintsAllowed {
 		return HintResult{}, ErrHintsDisabled
 	}
+	if pre.VsAI {
+		// vs_ai: unlimited and wallet-free, but idle-gated from the server clock — and counted toward
+		// no hint statistic. Enforce the gate (the client normally pre-gates from the view's
+		// HintUnlockLeftSeconds; this is the authoritative backstop), then serve the top move without
+		// touching the allowance, the wallet or hints_used.
+		if hintUnlockLeftSeconds(pre, seat, svc.clock()) > 0 {
+			return HintResult{}, ErrHintLocked
+		}
+		unlock := svc.locks.lock(gameID)
+		defer unlock()
+		g, err := svc.liveGame(ctx, pre)
+		if err != nil {
+			return HintResult{}, err
+		}
+		move, ok := g.HintView()
+		if !ok {
+			return HintResult{}, ErrNoHintAvailable
+		}
+		return HintResult{Move: move}, nil
+	}
 	acc, err := svc.accounts.GetByID(ctx, accountID)
 	if err != nil {
 		return HintResult{}, err
@@ -1208,6 +1249,8 @@ func (svc *Service) GameState(ctx context.Context, gameID, accountID uuid.UUID)
 		BagLen:         g.BagLen(),
 		HintsRemaining: hintsRemaining(pre.HintsPerPlayer, pre.Seats[seat].HintsUsed, acc.HintBalance),
 		WalletBalance:  acc.HintBalance,
+		// vs_ai idle-hint gate (seconds left; 0 for a human game / first move / not your turn).
+		HintUnlockLeftSeconds: hintUnlockLeftSeconds(pre, seat, svc.clock()),
 	}, nil
 }