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/inttest/robot_test.go

@@ -207,8 +207,8 @@ func TestMatchmakerSubstitutesRobotEndToEnd(t *testing.T) {
 	}
 }
 
-// TestRobotProactiveNudge checks the robot nudges the human after the idle
-// threshold on the human's turn.
+// TestRobotProactiveNudge checks the robot's lengthening proactive-nudge schedule on the
+// human's turn: nothing before the ~60-90 min first gap, exactly one once it has elapsed.
 func TestRobotProactiveNudge(t *testing.T) {
 	ctx := context.Background()
 	svc := newGameService()
@@ -232,14 +232,18 @@ func TestRobotProactiveNudge(t *testing.T) {
 		t.Fatalf("create: %v", err)
 	}
 
-	// Midnight start, driven 13 hours later (>12h idle) at a daytime hour awake for
-	// every drift.
-	start := time.Date(2024, 1, 1, 0, 0, 0, 0, time.UTC)
+	// A daytime turn start (the robot is awake for every ±3h drift between 07:30 and 12:00). No
+	// nudge before the 60-min floor of the first gap; exactly one once past its 90-min ceiling.
+	start := time.Date(2024, 1, 1, 10, 0, 0, 0, time.UTC)
 	setTurnStarted(t, g.ID, start)
-	robots.Drive(ctx, start.Add(13*time.Hour))
 
+	robots.Drive(ctx, start.Add(30*time.Minute))
+	if n := countNudges(t, g.ID, robotID); n != 0 {
+		t.Errorf("robot nudges = %d at 30m idle, want 0 (before the first gap)", n)
+	}
+	robots.Drive(ctx, start.Add(2*time.Hour))
 	if n := countNudges(t, g.ID, robotID); n != 1 {
-		t.Errorf("robot nudges = %d, want 1 after 13h idle on the human's turn", n)
+		t.Errorf("robot nudges = %d at 2h idle, want 1 (after the first gap)", n)
 	}
 }
 

backend/internal/inttest/social_test.go

@@ -502,6 +502,32 @@ func TestRespondPublishesToRequester(t *testing.T) {
 	}
 }
 
+// TestNudgeRoutedByGameLanguage checks a nudge's out-of-app push carries the game's language, so
+// it is delivered by the game's bot rather than the recipient's last-login bot (Stage 17).
+func TestNudgeRoutedByGameLanguage(t *testing.T) {
+	ctx := context.Background()
+	svc := newSocialService()
+	pub := &capturePublisher{}
+	svc.SetNotifier(pub)
+
+	gameID, seats := newGameWithSeats(t, 2) // an English game; seat 0 is to move
+	if _, err := svc.Nudge(ctx, gameID, seats[1]); err != nil {
+		t.Fatalf("nudge: %v", err)
+	}
+	found := false
+	for _, in := range pub.intents {
+		if in.Kind == notify.KindNudge {
+			found = true
+			if in.Language != "en" {
+				t.Errorf("nudge language = %q, want en (the game's language)", in.Language)
+			}
+		}
+	}
+	if !found {
+		t.Fatal("no nudge intent published")
+	}
+}
+
 // TestAdminListMessages checks the admin moderation list (Stage 17): real messages only
 // (nudges excluded), the game / sender pins, the sender glob masks, and the source label.
 func TestAdminListMessages(t *testing.T) {