Stage 17 round 5 (L2): robot play-to-win intent + next-move ETA in the admin game card

The admin game detail now shows, per robot seat, the game's deterministic play-to-win
decision (from the bag seed) and — while it is that robot's turn — its scheduled next-move
ETA (sampled think-time delay, deferred past the sleep window), plus a caption with the
~40% global target. Wiring: robot.PlayToWin/NextMoveAt/PlayToWinTargetPercent exports,
account.IsRobot, game RobotSchedule (seed + turn-start). Tests: NextMoveAt invariants
(never early, never in the sleep window), PlayToWin export, and an admin render integration
test asserting the intent + ETA + target appear.

backend/internal/game/store.go

@@ -651,6 +651,24 @@ func (s *Store) GameSeed(ctx context.Context, id uuid.UUID) (int64, error) {
 	return row.Seed, nil
 }
 
+// RobotSchedule returns a game's bag seed and current turn-start time. The admin console
+// combines them with the robot strategy to show a robot seat's play-to-win intent and its
+// next-move ETA. Both are server-only state, never part of the public game view.
+func (s *Store) RobotSchedule(ctx context.Context, id uuid.UUID) (seed int64, turnStartedAt time.Time, err error) {
+	stmt := postgres.SELECT(table.Games.Seed, table.Games.TurnStartedAt).
+		FROM(table.Games).
+		WHERE(table.Games.GameID.EQ(postgres.UUID(id))).
+		LIMIT(1)
+	var row model.Games
+	if err := stmt.QueryContext(ctx, s.db, &row); err != nil {
+		if errors.Is(err, qrm.ErrNoRows) {
+			return 0, time.Time{}, ErrNotFound
+		}
+		return 0, time.Time{}, fmt.Errorf("game: get schedule %s: %w", id, err)
+	}
+	return row.Seed, row.TurnStartedAt, nil
+}
+
 // projectGame builds a Game from a games row and its ordered seat rows.
 func projectGame(g model.Games, seats []model.GamePlayers) (Game, error) {
 	variant, err := engine.ParseVariant(g.Variant)