feat: sparser robot nudges, typed unread badge, lobby unread bump

Three owner-requested polish changes:

- robot: replace the lengthening 60-90 min -> 6 h proactive-nudge ramp with a
  flat uniform 9-12 h wait before every nudge; the existing sleep-window gate
  still skips and defers a nudge that would land in the robot's night.
- ui: colour the lobby/in-game unread dot by type -- the regular danger colour
  when a chat message is unread, a softer amber (--warn) when only nudges are.
  Adds a per-viewer unread_messages flag (chat_messages.kind='message') across
  the backend DTO, FlatBuffers wire, gateway transcode and the UI store.
- ui: float games with any unread notification to the top of the lobby's
  your-turn and opponent-turn sections (finished keeps its order), reusing the
  existing unread_chat flag.

Docs (ARCHITECTURE 7, FUNCTIONAL + _ru) updated. No DB migration; the new wire
field is backward-compatible.

backend/internal/robot/strategy_test.go

@@ -315,13 +315,13 @@ func TestDeviatesDistribution(t *testing.T) {
 // 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)
+		if first := proactiveNudgeGap(0, seed); first < 9*time.Hour || first > 12*time.Hour {
+			t.Fatalf("first gap %s out of [9h,12h] 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 g < 9*time.Hour || g > 12*time.Hour {
+				t.Fatalf("gap %s out of [9h,12h] 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)
@@ -337,8 +337,16 @@ func TestProactiveNudgeGap(t *testing.T) {
 		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)
+	// The window is flat: the gap distribution does not lengthen with idle time, so the median
+	// stays near the band centre (10.5 h) and barely moves between a fresh turn and a long-idle one.
+	early, late := median(0), median(12*time.Hour)
+	for _, m := range []float64{early, late} {
+		if m < 9*60 || m > 12*60 {
+			t.Fatalf("median gap %.0f min out of [540,720]", m)
+		}
+	}
+	if diff := math.Abs(early - late); diff > 30 {
+		t.Errorf("median gap should not shift with idle (flat window): idle0=%.0f idle12h=%.0f", early, late)
 	}
 }