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/game/Game.svelte

@@ -23,7 +23,7 @@
   import { centre, premiumGrid } from '../lib/premiums';
   import { variantNameKey, usesStarBlank, BLANK_STAR } from '../lib/variants';
   import { alphabetLetters, hasAlphabet } from '../lib/alphabet';
-  import { hintsLeft, hintGateRemainingMs, hintLockMinutes } from '../lib/hints';
+  import { hintsLeft, hintGateRemainingMs, hintLockMinutes, HINT_GATE_MS } from '../lib/hints';
   import { downloadUrl, shareOrDownloadGcg, shareUrlAsFile } from '../lib/share';
   import { insideVK, vkAndroidWebView, vkCopyText, vkDownloadFile, vkPlatform, vkShowImages } from '../lib/vk';
   import { getCachedGame, setCachedGame, setCachedDraft, type CachedGame } from '../lib/gamecache';
@@ -168,19 +168,30 @@
   // hint was spent in another game (see lib/hints).
   const hintCount = $derived(hintsLeft(view, app.profile?.hintBalance ?? 0));
   // A vs_ai game's hint is unlimited and wallet-free, but idle-gated (an anti-frustration aid: it
-  // unlocks only after the player has been stuck ~30 min on a turn). The unlock is a persisted
-  // wall-clock instant carried on the game view (`hintUnlockAtMs`, stamped by the source off the
-  // robot's reply and kept fresh by the move delta), so the wait survives leaving and reopening the
-  // app. It is sanitised on read (source + lib/hints cap it at now + the window) so a device clock
-  // set BACK cannot freeze the gate; a clock set forward just opens the hint early, harmless for a
-  // solo game. `now` ticks so the 🔒 lifts live at the mark. 0 = open (the human's first move).
-  let now = $state(Date.now());
-  const hintUnlockAt = $derived(view?.game.vsAi ? (view.hintUnlockAtMs ?? 0) : 0);
-  const hintRemaining = $derived(hintGateRemainingMs(hintUnlockAt, now));
+  // unlocks only after the player has been stuck ~30 min on a turn). The source (the SERVER clock
+  // online, the device clock offline) tells us the SECONDS LEFT (view.hintUnlockLeftSeconds); we
+  // anchor a MONOTONIC countdown (performance.now()) to it, so a client clock change cannot skew it.
+  // A fresh value arrives on load (armHintGate below); when the robot moves the wait is the full
+  // window. `monoNow` ticks so the 🔒 lifts live at the mark.
+  let hintGateStart = $state<number | null>(null); // performance.now() at the last anchor; null = open
+  let gateLeftMs = $state(0);
+  let monoNow = $state(performance.now());
+  const hintRemaining = $derived(hintGateRemainingMs(hintGateStart, gateLeftMs, monoNow));
   const hintGated = $derived(hintRemaining > 0);
+  // Anchor the countdown to a freshly received seconds-left (or open the gate). Called on every state
+  // load from the view, and on the robot's move to the full window (the robot just moved).
+  function armHintGate(leftSeconds: number): void {
+    if (leftSeconds > 0) {
+      hintGateStart = performance.now();
+      gateLeftMs = leftSeconds * 1000;
+    } else {
+      hintGateStart = null;
+      gateLeftMs = 0;
+    }
+  }
   $effect(() => {
     if (!view?.game.vsAi) return;
-    const iv = setInterval(() => (now = Date.now()), 10_000);
+    const iv = setInterval(() => (monoNow = performance.now()), 10_000);
     return () => clearInterval(iv);
   });
   // RACK_SIZE mirrors the engine's rules.RackSize (7 for every current variant). The exchange
@@ -225,6 +236,8 @@
         source.draftGet(id).catch(() => ''),
       ]);
       view = st;
+      // Anchor the vs_ai idle-hint countdown to the freshly fetched seconds-left (0 = open / non-vs_ai).
+      armHintGate(st.game.vsAi ? (st.hintUnlockLeftSeconds ?? 0) : 0);
       syncWallet(st.walletBalance);
       // Seed the unread flag from the authoritative state (the live stream only raises it).
       seedChatUnread(id, st.game.unreadChat, st.game.unreadMessages);
@@ -345,7 +358,11 @@
         // While composing, reload so a draft overlapping the new move is reconciled; otherwise apply
         // the move as a delta with no fetch.
         if (placement.pending.length > 0) void load();
-        else applyDelta(applyMoveDelta(cacheSnapshot(), { move: e.move, game: e.game, bagLen: e.bagLen, hintUnlockAtMs: e.hintUnlockAtMs }));
+        else {
+          applyDelta(applyMoveDelta(cacheSnapshot(), { move: e.move, game: e.game, bagLen: e.bagLen }));
+          // The robot just moved (vs_ai) → the human's turn begins with the full idle window.
+          if (view?.game.vsAi) armHintGate(HINT_GATE_MS / 1000);
+        }
       } else if (e.kind === 'your_turn' && e.gameId === id) {
         // The opponent_moved delta carries the new state; your_turn only confirms the turn. Refetch
         // only if we missed the move (our cached count trails the event's).
@@ -873,7 +890,7 @@
     // vs_ai: the idle gate replaces the wallet. While it is still closed, a tap only explains when the
     // hint unlocks (no wallet is ever spent) — matching the 🔒 badge. Measured fresh at tap time.
     if (view?.game.vsAi) {
-      const remaining = hintGateRemainingMs(hintUnlockAt, Date.now());
+      const remaining = hintGateRemainingMs(hintGateStart, gateLeftMs, performance.now());
       if (remaining > 0) {
         showToast(t('game.hintLockedIn', { n: hintLockMinutes(remaining) }), 'info');
         return;
@@ -915,6 +932,12 @@
       // The backend does not spend a hint when there is no move.
       if (e instanceof GatewayError && e.code === 'no_hint_available') {
         showToast(t('game.noHintOptions'), 'info');
+      } else if (e instanceof GatewayError && e.code === 'hint_locked') {
+        // The server-clock backstop refused a vs_ai hint the client thought was open (a rare client
+        // clock desync): re-sync the countdown from a fresh fetch and explain the gate.
+        await load();
+        const left = hintGateRemainingMs(hintGateStart, gateLeftMs, performance.now());
+        showToast(t('game.hintLockedIn', { n: hintLockMinutes(left || 60_000) }), 'info');
       } else {
         handleError(e);
       }