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.go

@@ -55,9 +55,16 @@ const (
 	// sleep window relative to the opponent's timezone, in hours.
 	sleepDriftHours = 3
 
-	// proactiveNudgeIdle is how long the robot waits on the human's turn before it
-	// proactively nudges (subject to the social once-per-hour-per-game limit).
-	proactiveNudgeIdle = 12 * time.Hour
+	// The robot proactively nudges the idle human on a lengthening, randomized schedule rather
+	// than an hourly stream: the first nudge lands ~60-90 min into the turn, and each subsequent
+	// gap grows toward 1-6 h the longer the wait drags on, so a long idle turn gets only a handful
+	// of increasingly-spaced reminders. The gap is a uniform sample in [nudgeGapFloorMinutes,
+	// ceil] minutes, where ceil ramps from nudgeGapFirstCeilMinutes to nudgeGapCeilMinutes over
+	// nudgeGapRamp of idle.
+	nudgeGapFloorMinutes     = 60.0
+	nudgeGapFirstCeilMinutes = 90.0
+	nudgeGapCeilMinutes      = 360.0
+	nudgeGapRamp             = 12 * time.Hour
 )
 
 // defaultBand is the target resulting score margin after the robot's move: when
@@ -181,6 +188,23 @@ func nudgeReplyDelay(seed int64, moveCount int) time.Duration {
 	return clampMinutes(lo + nudgeReplySpreadMinutes*u)
 }
 
+// proactiveNudgeGap is the randomized wait before the next proactive nudge, given how long the
+// human had already been idle at the previous nudge (refIdle; 0 for the first nudge of the turn).
+// It is a uniform sample in [nudgeGapFloorMinutes, ceil] minutes, where ceil ramps from
+// nudgeGapFirstCeilMinutes (a ~60-90 min first gap) up to nudgeGapCeilMinutes (a 1-6 h gap) as
+// refIdle reaches nudgeGapRamp — so the reminders space out the longer the turn is neglected. It
+// is deterministic per (seed, refIdle), so the driver computes the same due time on every scan.
+func proactiveNudgeGap(refIdle time.Duration, seed int64) time.Duration {
+	f := float64(refIdle) / float64(nudgeGapRamp)
+	if f > 1 {
+		f = 1
+	}
+	ceil := nudgeGapFirstCeilMinutes + (nudgeGapCeilMinutes-nudgeGapFirstCeilMinutes)*f
+	u := unitFloat(mix(seed, "pnudge", int(refIdle/(30*time.Minute))))
+	mins := nudgeGapFloorMinutes + (ceil-nudgeGapFloorMinutes)*u
+	return time.Duration(mins * float64(time.Minute))
+}
+
 // clampMinutes converts a minute count to a duration, clamping it to the hard delay
 // bounds so an out-of-range band can never produce an absurd think time.
 func clampMinutes(mins float64) time.Duration {