wallpaper: persist cycling state to disk

fixes #1010

core/internal/server/wallpaper/manager_test.go

@@ -83,8 +83,81 @@ func TestActiveSchedules(t *testing.T) {
 	}
 }
 
+func TestPrevDailyTime(t *testing.T) {
+	now := time.Date(2026, 6, 30, 12, 0, 0, 0, time.UTC)
+
+	earlier, ok := prevDailyTime(now, "06:00")
+	if !ok || !earlier.Equal(time.Date(2026, 6, 30, 6, 0, 0, 0, time.UTC)) {
+		t.Errorf("prev (today) = %v, %v", earlier, ok)
+	}
+
+	later, ok := prevDailyTime(now, "18:00")
+	if !ok || !later.Equal(time.Date(2026, 6, 29, 18, 0, 0, 0, time.UTC)) {
+		t.Errorf("prev (yesterday) = %v, %v", later, ok)
+	}
+
+	if _, ok := prevDailyTime(now, "bad"); ok {
+		t.Error("prev (invalid) should not be ok")
+	}
+}
+
+func TestTimeScheduleCatchUpAfterRestart(t *testing.T) {
+	origDelay := catchUpDelay
+	catchUpDelay = 100 * time.Millisecond
+	defer func() { catchUpDelay = origDelay }()
+
+	now := time.Now()
+	hhmm := now.Add(-2 * time.Hour).Format("15:04")
+	seed := map[string]time.Time{"": now.Add(-26 * time.Hour)}
+
+	m := newManager(seed, func(map[string]time.Time) {})
+	defer m.Close()
+
+	sub := m.Subscribe("test")
+	defer m.Unsubscribe("test")
+
+	m.SetConfig(Config{Global: ScheduleConfig{Enabled: true, Mode: "time", Time: hhmm}})
+
+	deadline := time.After(3 * time.Second)
+	for {
+		select {
+		case state := <-sub:
+			if state.CycleSeq > 0 && state.Target == "" {
+				return
+			}
+		case <-deadline:
+			t.Fatal("missed time schedule did not catch up within 3s")
+		}
+	}
+}
+
+func TestTimeScheduleNoFireOnFirstEnable(t *testing.T) {
+	now := time.Now()
+	hhmm := now.Add(-2 * time.Hour).Format("15:04")
+
+	m := newManager(map[string]time.Time{}, func(map[string]time.Time) {})
+	defer m.Close()
+
+	sub := m.Subscribe("test")
+	defer m.Unsubscribe("test")
+
+	m.SetConfig(Config{Global: ScheduleConfig{Enabled: true, Mode: "time", Time: hhmm}})
+
+	deadline := time.After(1500 * time.Millisecond)
+	for {
+		select {
+		case state := <-sub:
+			if state.CycleSeq > 0 {
+				t.Fatal("first enable must not fire immediately")
+			}
+		case <-deadline:
+			return
+		}
+	}
+}
+
 func TestSchedulerEmitsCycle(t *testing.T) {
-	m := NewManager()
+	m := newManager(map[string]time.Time{}, func(map[string]time.Time) {})
 	defer m.Close()
 
 	sub := m.Subscribe("test")