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/server/dto.go

@@ -163,13 +163,16 @@ type alphabetEntryDTO struct {
 // stateDTO is a player's view of a game. Rack carries wire alphabet indices (a
 // blank is engine.BlankIndex). Alphabet is present only when the request asked for it.
 type stateDTO struct {
-	Game           gameDTO            `json:"game"`
-	Seat           int                `json:"seat"`
-	Rack           []int              `json:"rack"`
-	BagLen         int                `json:"bag_len"`
-	HintsRemaining int                `json:"hints_remaining"`
-	WalletBalance  int                `json:"wallet_balance"`
-	Alphabet       []alphabetEntryDTO `json:"alphabet,omitempty"`
+	Game           gameDTO `json:"game"`
+	Seat           int     `json:"seat"`
+	Rack           []int   `json:"rack"`
+	BagLen         int     `json:"bag_len"`
+	HintsRemaining int     `json:"hints_remaining"`
+	WalletBalance  int     `json:"wallet_balance"`
+	// HintUnlockLeftSeconds is the vs_ai idle-hint gate: seconds until the hint unlocks (0 for a human
+	// game / first move / not your turn). The client anchors a monotonic countdown to it.
+	HintUnlockLeftSeconds int                `json:"hint_unlock_left_seconds"`
+	Alphabet              []alphabetEntryDTO `json:"alphabet,omitempty"`
 }
 
 // matchDTO reports whether the caller has been paired into a game.
@@ -315,12 +318,13 @@ func stateDTOFrom(v game.StateView, includeAlphabet bool) (stateDTO, error) {
 		return stateDTO{}, err
 	}
 	dto := stateDTO{
-		Game:           gameDTOFromGame(v.Game),
-		Seat:           v.Seat,
-		Rack:           rack,
-		BagLen:         v.BagLen,
-		HintsRemaining: v.HintsRemaining,
-		WalletBalance:  v.WalletBalance,
+		Game:                  gameDTOFromGame(v.Game),
+		Seat:                  v.Seat,
+		Rack:                  rack,
+		BagLen:                v.BagLen,
+		HintsRemaining:        v.HintsRemaining,
+		WalletBalance:         v.WalletBalance,
+		HintUnlockLeftSeconds: v.HintUnlockLeftSeconds,
 	}
 	if includeAlphabet {
 		tab, err := engine.AlphabetTable(v.Game.Variant)