feat(chat): limit in-game chat to one message per turn

Enforce one chat message per turn on both ends. The backend rejects a
second message in the same turn (ErrChatAlreadySentThisTurn -> 409
chat_already_sent), keyed on the move-driven turn start (turn_started_at).
The UI derives "already wrote this turn" from the message list against
GameView.lastActivityUnix (no counter, survives reopening, resets on turn
change), hides the field behind a short caption once the limit is reached,
and now reloads the game state on turn/game-state events so the toggle and
the limit follow the live game. Enter is gated on !busy to avoid a
double-send in the in-flight window.

Backend: new game.TurnStartedAt; social GameReader gains it; PostMessage
enforces the limit reusing lastMessageAt. UI: new lib/chatlimit.ts pure
logic + unit tests; Chat/ChatScreen wiring; chat.sentThisTurn and
error.chat_already_sent i18n (en/ru); extended chat e2e. Docs: FUNCTIONAL
(+ru), ARCHITECTURE, UI_DESIGN.

backend/internal/game/service.go

@@ -408,6 +408,12 @@ func (svc *Service) LastMoveAt(ctx context.Context, gameID, accountID uuid.UUID)
 	return svc.store.LastMoveAt(ctx, gameID, accountID)
 }
 
+// TurnStartedAt returns the start time of a game's current turn. The social service uses
+// it as the per-turn boundary for the one-chat-message-per-turn limit.
+func (svc *Service) TurnStartedAt(ctx context.Context, gameID uuid.UUID) (time.Time, error) {
+	return svc.store.TurnStartedAt(ctx, gameID)
+}
+
 // transition validates the actor and turn, applies op under the per-game lock and
 // commits the result.
 func (svc *Service) transition(ctx context.Context, gameID, accountID uuid.UUID, op engineOp) (MoveResult, error) {

backend/internal/game/store.go

@@ -968,6 +968,19 @@ func (s *Store) LastMoveAt(ctx context.Context, gameID, accountID uuid.UUID) (ti
 	return at.Time, true, nil
 }
 
+// TurnStartedAt returns the start time of a game's current turn (games.turn_started_at).
+// The social service uses it as the per-turn boundary for the one-chat-message-per-turn
+// limit; it advances only on a move, never on chat or a nudge.
+func (s *Store) TurnStartedAt(ctx context.Context, gameID uuid.UUID) (time.Time, error) {
+	var at time.Time
+	err := s.db.QueryRowContext(ctx,
+		`SELECT turn_started_at FROM backend.games WHERE game_id = $1`, gameID).Scan(&at)
+	if err != nil {
+		return time.Time{}, fmt.Errorf("game: turn started at %s: %w", gameID, err)
+	}
+	return at, 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.