Q: My neighbor above me has kids that run & stomp all day & late into the evening. It starts early in the morning, even as early as 4 a.m. What can I do or say that won't cause animosity?
A: Oof, I feel your pain, but I think you're on the right track by saying you don't want to cause animosity. Friendly communication is always a good start as most people want to be considerate to those that are nice. The last thing you want is a situation where they are offended and decide to make more noise on purpose.
First of all, consider the fact that they may not be aware of the noise level, or how it's affecting you. You might want to bake a batch of cookies and go up there to deliver them and find out what's really going in their household. That may give you some clues before you decide how to handle the situation. Polite chit chat - like letting them know you're there if they need anything - may naturally lead to talk about the noise.
By establishing friendly communication you can then initiate seemingly innocuous conversations like, "Wow, Johnny was up early today, wasn't he?" Or "Billy sure had a late night!" They may be surprised that you can actually hear the noise and be horrified that you can track their moves.
If that doesn't work, let the property management company know of your issue and let them handle it on your behalf. On your end, you may want to invest in a white noise generator (or just download free from the internet on websites like simplynoise.com) , or decorate with some wall tapestries to absorb and muffle the noise.