Advantages of adaboost over gentleboost in applications, or vice versa?
Genetellboost is developed to cope with labeling noise, i.e., when some samples are wrongly labeled. Though, it wasn't so successful in that respect, from my experience it is quite good in handling noisy features which are kind of outliers in dataset. That being said, there is no proof for neither of the things that you asked (nor the things that you mentioned are true). In fact, we know that these algorithms are so similar that it is practically impossible to predict which one works better on which dataset. You have to test it and that's it.