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.

ui/src/lib/localgame/source.ts

@@ -290,10 +290,9 @@ export class LocalSource implements GameLoopSource {
   private emitRobot(entry: Live, moves: LocalMove[]): void {
     const set = this.listeners.get(entry.record.id);
     if (!set) return;
-    const hintUnlockAtMs = this.hintUnlock(entry);
     for (const m of moves) {
       const game = this.gameView(entry);
-      const ev: PushEvent = { kind: 'opponent_moved', gameId: entry.record.id, move: this.moveRecord(entry.record.variant, m), game, bagLen: entry.game.bagLength, hintUnlockAtMs };
+      const ev: PushEvent = { kind: 'opponent_moved', gameId: entry.record.id, move: this.moveRecord(entry.record.variant, m), game, bagLen: entry.game.bagLength };
       for (const cb of set) cb(ev);
     }
     if (entry.game.isOver) {
@@ -342,11 +341,14 @@ export class LocalSource implements GameLoopSource {
 
   // --- shape builders --------------------------------------------------------
 
-  // The vs_ai idle-hint unlock instant, sanitised on read: capped at now + the window so a device
-  // clock set back cannot push it far ahead and freeze the gate (the client counts down from here).
-  // 0 while open (a human-first opening, no robot move yet).
-  private hintUnlock(entry: Live): number {
-    return entry.record.hintUnlockAtMs ? Math.min(entry.record.hintUnlockAtMs, Date.now() + HINT_GATE_MS) : 0;
+  // The vs_ai idle-hint seconds left, from the stored unlock instant (device wall clock, persisted
+  // for the offline record) minus now — capped at the window so a device clock set back cannot push
+  // it far ahead and freeze the gate, ceiled to whole seconds and floored at 0. The client anchors a
+  // monotonic countdown to this; 0 while open (a human-first opening, no robot move yet).
+  private hintUnlockLeft(entry: Live): number {
+    if (!entry.record.hintUnlockAtMs) return 0;
+    const leftMs = Math.min(HINT_GATE_MS, entry.record.hintUnlockAtMs - Date.now());
+    return leftMs > 0 ? Math.ceil(leftMs / 1000) : 0;
   }
 
   private stateView(entry: Live): StateView {
@@ -359,7 +361,7 @@ export class LocalSource implements GameLoopSource {
       bagLen: entry.game.bagLength,
       hintsRemaining: 1,
       walletBalance: 0,
-      hintUnlockAtMs: this.hintUnlock(entry),
+      hintUnlockLeftSeconds: this.hintUnlockLeft(entry),
     };
   }