Stage 17: fix the robot-nudge frequency + per-game push language

Two owner-reported defects from a live contour game.

A. Frequency: the robot's proactive nudge fired hourly for 12h+ (a 12h idle threshold
   then the 1h cooldown, uncapped). Replaced with a lengthening, randomized schedule
   (proactiveNudgeGap): the first nudge ~60-90 min into the human's turn, each later gap
   growing toward 1-6h (uniform sample in [60min, ceil], ceil ramping 90min->6h over 12h
   of idle, measured from the previous nudge), so a long wait gets a handful of
   increasingly-spaced reminders instead of a stream.

B. Language: out-of-app push routed by the recipient's GLOBAL service_language
   (last-login-wins), so after re-logging via the RU bot an English game's nudges came
   from the RU bot. Now a game push (your_turn, game_over, nudge, match_found) carries
   the game's own language (engine.Variant.Language) on push.Event, and the gateway
   routes by it (falling back to service_language for non-game pushes). The New-Game
   variant-gating guarantees the game's bot is one the player has started, so delivery is
   never blocked.

Tests: proactiveNudgeGap unit + retimed TestRobotProactiveNudge; TestVariantLanguage;
emit your_turn/game_over language; TestNudgeRoutedByGameLanguage integration. Docs:
ARCHITECTURE (§7 nudge, §10/§13 routing), FUNCTIONAL (+ _ru), PLAN tracker.

backend/internal/game/emit_test.go

@@ -68,6 +68,10 @@ func TestEmitMoveNotifiesActor(t *testing.T) {
 	if got := string(yt.ScoreLine()); got != "13:19" { // seat 1 (recipient) first, then seat 0
 		t.Errorf("your_turn score_line = %q, want 13:19", got)
 	}
+	// Routed out-of-app by the game's language (the default Variant is English).
+	if yourTurn.Language != "en" {
+		t.Errorf("your_turn language = %q, want en", yourTurn.Language)
+	}
 }
 
 // TestEmitMoveAnnouncesGameOver checks the closing move sends a game_over push to every seat,
@@ -102,4 +106,7 @@ func TestEmitMoveAnnouncesGameOver(t *testing.T) {
 	if string(l.Result()) != "lost" || string(l.ScoreLine()) != "95:120" {
 		t.Errorf("loser game_over = %q / %q, want lost / 95:120", l.Result(), l.ScoreLine())
 	}
+	if over[winner].Language != "en" || over[loser].Language != "en" {
+		t.Errorf("game_over languages = %q/%q, want en/en", over[winner].Language, over[loser].Language)
+	}
 }

backend/internal/game/service.go

@@ -222,6 +222,17 @@ func (svc *Service) GameVariant(ctx context.Context, gameID uuid.UUID) (engine.V
 	return svc.store.GetGameVariant(ctx, gameID)
 }
 
+// GameLanguage returns the game's language tag ("en"/"ru"), derived from its variant, so a
+// game push routes out-of-app to the game's own bot rather than the recipient's last-login bot
+// (Stage 17).
+func (svc *Service) GameLanguage(ctx context.Context, gameID uuid.UUID) (string, error) {
+	v, err := svc.GameVariant(ctx, gameID)
+	if err != nil {
+		return "", err
+	}
+	return v.Language(), nil
+}
+
 // RobotSchedule returns a game's bag seed and turn-start time, for the admin console's
 // robot-schedule panel (the deterministic play-to-win intent and next-move ETA).
 func (svc *Service) RobotSchedule(ctx context.Context, gameID uuid.UUID) (seed int64, turnStartedAt time.Time, err error) {
@@ -367,6 +378,9 @@ func (svc *Service) emitMove(ctx context.Context, post Game, rec engine.MoveReco
 	for _, s := range post.Seats {
 		intents = append(intents, notify.OpponentMoved(s.AccountID, post.ID, rec.Player, rec.Action.String(), rec.Score, rec.Total))
 	}
+	// Game pushes are routed out-of-app by the game's own language, not the recipient's
+	// last-login bot (Stage 17).
+	lang := post.Variant.Language()
 	switch post.Status {
 	case StatusActive:
 		if next, ok := seatAccount(post.Seats, post.ToMove); ok {
@@ -377,14 +391,18 @@ func (svc *Service) emitMove(ctx context.Context, post Game, rec engine.MoveReco
 				word = rec.Words[0]
 			}
 			opponent := svc.displayName(ctx, post.Seats, rec.Player)
-			intents = append(intents, notify.YourTurn(next, post.ID, deadline, opponent, action, word, scoreLine(post, post.ToMove)))
+			yourTurn := notify.YourTurn(next, post.ID, deadline, opponent, action, word, scoreLine(post, post.ToMove))
+			yourTurn.Language = lang
+			intents = append(intents, yourTurn)
 		}
 	case StatusFinished:
 		// The game just ended (any path: a closing play, all-pass, resign or timeout). Tell every
 		// seat, each with their own perspective + recipient-first score, so an offline player gets
 		// an out-of-app "game over" push (online players take it from the in-app refresh).
 		for _, s := range post.Seats {
-			intents = append(intents, notify.GameOver(s.AccountID, post.ID, seatResult(post.Seats, s.Seat), scoreLine(post, s.Seat)))
+			over := notify.GameOver(s.AccountID, post.ID, seatResult(post.Seats, s.Seat), scoreLine(post, s.Seat))
+			over.Language = lang
+			intents = append(intents, over)
 		}
 	}
 	svc.pub.Publish(intents...)