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/robot/strategy_test.go

@@ -238,6 +238,38 @@ func TestPlayToWinExport(t *testing.T) {
 	}
 }
 
+// TestProactiveNudgeGap checks the proactive-nudge schedule: the first gap (refIdle 0) is
+// ~60-90 min, every gap stays within [60 min, 6 h] and is deterministic, and the gap lengthens
+// as the idle grows (the median at 12 h idle exceeds the median at the start).
+func TestProactiveNudgeGap(t *testing.T) {
+	for seed := int64(1); seed <= 1000; seed++ {
+		if first := proactiveNudgeGap(0, seed); first < 60*time.Minute || first > 90*time.Minute {
+			t.Fatalf("first gap %s out of [60m,90m] for seed %d", first, seed)
+		}
+		for _, idle := range []time.Duration{0, time.Hour, 3 * time.Hour, 6 * time.Hour, 12 * time.Hour, 24 * time.Hour} {
+			g := proactiveNudgeGap(idle, seed)
+			if g < 60*time.Minute || g > 6*time.Hour {
+				t.Fatalf("gap %s out of [60m,6h] for seed %d idle %s", g, seed, idle)
+			}
+			if proactiveNudgeGap(idle, seed) != g {
+				t.Fatalf("gap not deterministic for seed %d idle %s", seed, idle)
+			}
+		}
+	}
+	median := func(idle time.Duration) float64 {
+		const n = 4000
+		xs := make([]float64, n)
+		for s := 0; s < n; s++ {
+			xs[s] = proactiveNudgeGap(idle, int64(s+1)).Minutes()
+		}
+		sort.Float64s(xs)
+		return xs[n/2]
+	}
+	if early, late := median(0), median(12*time.Hour); early >= late {
+		t.Errorf("median gap should grow with idle: idle0=%.0f idle12h=%.0f", early, late)
+	}
+}
+
 // plays builds candidate plays carrying only the given scores (ranked as passed).
 func plays(scores ...int) []engine.MoveRecord {
 	out := make([]engine.MoveRecord, len(scores))